P. 452 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

fsovd 45101*

Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))

Theorem

fsovrfovd 45102*

The operator which gives a 1-to-1 a mapping to a subset and a reverse mapping from elements can be composed from the operator which gives a 1-to-1 mapping between relations and functions to subsets and the converse operator. (Contributed by RP, 15-May-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑢 ∈ 𝑎 ↦ {𝑣 ∈ 𝑏 ∣ 𝑢𝑟𝑣})))    &   ⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ ^◡𝑠))    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵) = ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ ^◡(𝐴𝑅𝐵))))

Theorem

fsovfvd 45103*

Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, when applied to function 𝐹. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝐴))    ⇒   ⊢ (𝜑 → (𝐺‘𝐹) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

Theorem

fsovfvfvd 45104*

Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, when applied to function 𝐹 and element 𝑌. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝐴))    &   ⊢ 𝐻 = (𝐺‘𝐹)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻‘𝑌) = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})

Theorem

fsovfd 45105*

The operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, gives a function between two sets of functions. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑_m 𝐴)⟶(𝒫 𝐴 ↑_m 𝐵))

Theorem

fsovcnvlem 45106*

The 𝑂 operator, which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ 𝐻 = (𝐵𝑂𝐴)    ⇒   ⊢ (𝜑 → (𝐻 ∘ 𝐺) = ( I ↾ (𝒫 𝐵 ↑_m 𝐴)))

Theorem

fsovcnvd 45107*

The value of the converse (𝐴𝑂𝐵) is (𝐵𝑂𝐴), where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ 𝐻 = (𝐵𝑂𝐴)    ⇒   ⊢ (𝜑 → ^◡𝐺 = 𝐻)

Theorem

fsovcnvfvd 45108*

The value of the converse of (𝐴𝑂𝐵), where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, evaluated at function 𝐹. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐴 ↑_m 𝐵))    ⇒   ⊢ (𝜑 → (^◡𝐺‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

Theorem

fsovf1od 45109*

The value of (𝐴𝑂𝐵) is a bijection, where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑_m 𝐴)–1-1-onto→(𝒫 𝐴 ↑_m 𝐵))

Theorem

dssmapfvd 45110*

Value of the duality operator for self-mappings of subsets of a base set, 𝐵. (Contributed by RP, 19-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))

Theorem

dssmapfv2d 45111*

Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹. (Contributed by RP, 19-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))    &   ⊢ 𝐺 = (𝐷‘𝐹)    ⇒   ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))

Theorem

dssmapfv3d 45112*

Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹 and subset 𝑆. (Contributed by RP, 19-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))    &   ⊢ 𝐺 = (𝐷‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ 𝑇 = (𝐺‘𝑆)    ⇒   ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))

Theorem

dssmapnvod 45113*

For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is its own inverse, an involution. (Contributed by RP, 20-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ^◡𝐷 = 𝐷)

Theorem

dssmapf1od 45114*

For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is one-to-one and onto. (Contributed by RP, 21-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑_m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

dssmap2d 45115*

For any base set 𝐵 the duality operator for self-mappings of subsets of that base set when composed with itself is the restricted identity operator. (Contributed by RP, 21-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑_m 𝒫 𝐵)))

21.40.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods

For any base set, 𝐵, an arbitrary mapping of subsets to subsets can be called a pseudoclosure (pseudointerior) function, 𝐾, with its dual of a pseudointerior (pseudoclosure), 𝐼, related by the involution in dssmapfvd 45110. As 𝐾 gains properties of the closure (interior) function of a topology on 𝐵, so does its dual gain corresponding properties of the interior (closure) function of that topology.

As (𝒫 𝐵 ↑_m 𝒫 𝐵) ≈ (𝒫 𝒫 𝐵 ↑_m 𝐵) there is also a natural isomorphism which maps from 𝐼 to 𝑁 (and likewise for 𝐾 and 𝑀, introduced below) which identically gains the properties of the neighborhood function of a topology (modified and restricted to operate on single points). A function dual to 𝑁, which Stadler and Stadler refer to as a convergent function, is represented by 𝑀 in this section.

Based on this and the early treatment of topology in Seifert and Threlfall, it seems reasonable to define a pseudotopology as defined in terms of its base set and one of these functions with theorems treating the equivalence of the other definitions and adding topological structure if enough properties hold true.

	Neighborhoods		Interior		Closure		Convergents	Theorems
Functions	𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)		𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)		𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)		𝑀 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)
Correspondences (assuming (𝑋 ∈ 𝐵 ∧ 𝑆 ∈ 𝒫 𝐵))	𝑆 ∈ (𝑁‘𝑋)	↔	𝑋 ∈ (𝐼‘𝑆)	↔	¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))	↔	¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑋)	ntrclselnel1 45150, ntrneiel 45174, neicvgel1 45212
	¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)	↔	¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆))	↔	𝑋 ∈ (𝐾‘𝑆)	↔	𝑆 ∈ (𝑀‘𝑋)
Neighborhoods	(𝑁‘𝑋)	=	{𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑠))}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}	ntrneifv3 45175, clsneifv3 45203, neicvgfv 45214
Interior	{𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)}	=	(𝐼‘𝑆)	=	(𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))	=	{𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑥)}	ntrneifv4 45178, ntrclsfv 45152, clsneifv4 45204
Closure	{𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}	=	(𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆)))	=	(𝐾‘𝑆)	=	{𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑀‘𝑥)}	clsneifv4 45204, ntrclsfv 45152, ntrneifv4 45178
Convergents	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑁‘𝑋)}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠))}	=	{𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐾‘𝑠)}	=	(𝑀‘𝑋)	neicvgfv 45214, clsneifv3 45203, ntrneifv3 45175

We have the following table of equivalences to axioms largely established by Kuratowski. In the formulas in this table, to reduce the width of the columns, if any of the variables 𝑥, 𝑠, or 𝑡 are used, then they are implicitly universally quantified and 𝑥 (respectively 𝑠 and 𝑡) ranges over 𝐵 (respectively 𝒫 𝐵 and 𝒫 𝐵).

Assuming a prefix of: ∀𝑥 ∈ 𝐵∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵	Neighborhoods	Interior	Closure	Convergents	Equivalence Theorems
K0' Neighborhoods are nonempty.	(𝑁‘𝑥) ≠ ∅	∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑢)	∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐾‘𝑢)	(𝑀‘𝑥) ≠ 𝒫 𝐵	ntrclsneine0 45158, ntrneineine0 45180, ntrneineine1 45181
KA' No neighborhood is equal to the full powerset.	(𝑁‘𝑥) ≠ 𝒫 𝐵	∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐼‘𝑢)	∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐾‘𝑢)	(𝑀‘𝑥) ≠ ∅	ntrclsneine0 45158, ntrneineine0 45180, ntrneineine1 45181
K0 Preservation of the Nullary Union of Closures	𝐵 ∈ (𝑁‘𝑥)	(𝐼‘𝐵) = 𝐵	(𝐾‘∅) = ∅	¬ ∅ ∈ (𝑀‘𝑥)	ntrclscls00 45159, ntrneicls00 45182, ntrneicls11 45183
KA Preservation of the Nullary Union of Interiors	¬ ∅ ∈ (𝑁‘𝑥)	(𝐼‘∅) = ∅	(𝐾‘𝐵) = 𝐵	𝐵 ∈ (𝑀‘𝑥)	ntrclscls00 45159, ntrneicls00 45182, ntrneicls11 45183
K1 Isotonic Montonic	((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))	(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) — or — ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∪ 𝑡)) — or — (𝐼‘(𝑠 ∩ 𝑡)) ⊆ ((𝐼‘𝑠) ∩ (𝐼‘𝑡))	(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) — or — ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) ⊆ (𝐾‘(𝑠 ∪ 𝑡)) — or — (𝐾‘(𝑠 ∩ 𝑡)) ⊆ ((𝐾‘𝑠) ∩ (𝐾‘𝑡))	((𝑠 ∈ (𝑀‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑀‘𝑥))	isotone1 45141, isotone2 45142, ntrclsiso 45160, ntrneiiso 45184
K2 Closure is Expansive	(𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠)	(𝐼‘𝑠) ⊆ 𝑠	𝑠 ⊆ (𝐾‘𝑠)	(𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑀‘𝑥))	ntrclsk2 45161, ntrneik2 45185, ntrneix2 45186
KB Non-disjoint Neighborhoods	((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ≠ ∅)	((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)	((𝑠 ∪ 𝑡) = 𝐵 → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = 𝐵)	((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥)))	ntrclskb 45162, ntrneikb 45187, ntrneixb 45188
K3 Closure is Sub-linear	((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥))	((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡))	(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))	((𝑠 ∪ 𝑡) ∈ (𝑀‘𝑥) → (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥)))	ntrclsk3 45163, ntrneik3 45189, ntrneix3 45190
K13 Closure is finitely linear	((𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)))	(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡))	(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))	((𝑠 ∪ 𝑡) ∈ (𝑀‘𝑥) ↔ (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥)))	ntrclsk13 45164, ntrneik13 45191, ntrneix13 45192
K4 Closure is idempotent	(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))	(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠)	(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)	(𝑠 ∈ (𝑀‘𝑥) ↔ ∃𝑢 ∈ (𝑀‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑀‘𝑦)))	ntrclsk4 45165, ntrneik4 45194

Using these properties as axiomic constraints on the functions, certain collections of them give rise to named spaces.

Space	Foundational Axioms	Derived Axioms	Theorems
Csázár Generalized Neighborhood Space	K2	KA', KA, KB	ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129
Min Strong Generalized Neighborhood Space	K2, K3	KA', KA, KB	ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129
Gniłka Extended Topology	K0', K1	K0	neik0pk1imk0 45140
Brissaud Space	K0, K2	K0', KA', KA, KB	neik0imk0p 45129, ntrk2imkb 45130, ntrkbimka 45131
Neighborhood Space	K0', K1, K2	K0, KA', KA, KB	neik0pk1imk0 45140, ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129
Davey and Priestley Intersection Structure	K1, K4
Moore Closure Space	K1, K2, K4	KA', KA, KB	ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129
Convex Closure Space	K0', K1, K2, K4	K0, KA', KA, KB	neik0pk1imk0 45140, ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129
Smyth Neighborhood Space	K0', K13	K0, K1, K3	neik0pk1imk0 45140, ntrk1k3eqk13 45143
Čech Closure Space Pretopological Space	K0', K2, K13	K0, K1, KA', KA, KB, K3	neik0pk1imk0 45140, ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129, ntrk1k3eqk13 45143
Topological Space	K0', K2, K13, K4	K0, K1, KA', KA, KB, K3	neik0pk1imk0 45140, ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129, ntrk1k3eqk13 45143
Alexandroff Space	K0', K2, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 45140, ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129, ntrk1k3eqk13 45143, TBD
Alexandroff Topological Space	K0', K2, K4, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 45140, ntrk2imkb 45130, ntrkbimka 45131, neik0imk0p 45129, ntrk1k3eqk13 45143, TBD

Theorem

or3or 45116

Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)

⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))

Theorem

andi3or 45117

Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)

⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)))

Theorem

uneqsn 45118

If a union of classes is equal to a singleton then at least one class is equal to the singleton while the other may be equal to the empty set. (Contributed by RP, 5-Jul-2021.)

⊢ ((𝐴 ∪ 𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))

Theorem

brfvimex 45119

If a binary relation holds and the relation is the value of a function, then the argument to that function is a set. (Contributed by RP, 22-May-2021.)

⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐹‘𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)

Theorem

brovmptimex 45120*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))

Theorem

brovmptimex1 45121*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)

Theorem

brovmptimex2 45122*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)

Theorem

brcoffn 45123

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 7-Jun-2021.)

⊢ (𝜑 → 𝐶 Fn 𝑌)    &   ⊢ (𝜑 → 𝐷:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐷(𝐷‘𝐴) ∧ (𝐷‘𝐴)𝐶𝐵))

Theorem

brcofffn 45124

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶 Fn 𝑍)    &   ⊢ (𝜑 → 𝐷:𝑌⟶𝑍)    &   ⊢ (𝜑 → 𝐸:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵))

Theorem

brco2f1o 45125

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)    &   ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)    ⇒   ⊢ (𝜑 → ((^◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(^◡𝐶‘𝐵)))

Theorem

brco3f1o 45126

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)    &   ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)    &   ⊢ (𝜑 → 𝐸:𝑊–1-1-onto→𝑋)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)    ⇒   ⊢ (𝜑 → ((^◡𝐶‘𝐵)𝐶𝐵 ∧ (^◡𝐷‘(^◡𝐶‘𝐵))𝐷(^◡𝐶‘𝐵) ∧ 𝐴𝐸(^◡𝐷‘(^◡𝐶‘𝐵))))

Theorem

ntrclsbex 45127

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then the base set exists. (Contributed by RP, 21-May-2021.)

⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)

Theorem

ntrclsrcomplex 45128

The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 25-Jun-2021.)

⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Theorem

neik0imk0p 45129

Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom implies KA' when considering the convergents. (Contributed by RP, 28-Jun-2021.)

⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)

Theorem

ntrk2imkb 45130*

If an interior function is contracting, the interiors of disjoint sets are disjoint. Kuratowski's K2 axiom implies KB. Interior version. (Contributed by RP, 9-Jun-2021.)

⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))

Theorem

ntrkbimka 45131*

If the interiors of disjoint sets are disjoint, then the interior of the empty set is the empty set. (Contributed by RP, 14-Jun-2021.)

⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → (𝐼‘∅) = ∅)

Theorem

ntrk0kbimka 45132*

If the interiors of disjoint sets are disjoint and the interior of the base set is the base set, then the interior of the empty set is the empty set. Obsolete version of ntrkbimka 45131. (Contributed by RP, 12-Jun-2021.)

⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) → (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘∅) = ∅))

Theorem

clsk3nimkb 45133*

If the base set is not empty, axiom K3 does not imply KB. A concrete example with a pseudo-closure function of 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) is given. (Contributed by RP, 16-Jun-2021.)

⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏))

Theorem

clsk1indlem0 45134

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ (𝐾‘∅) = ∅

Theorem

clsk1indlem2 45135*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o𝑠 ⊆ (𝐾‘𝑠)

Theorem

clsk1indlem3 45136*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K3 property of being sub-linear. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o∀𝑡 ∈ 𝒫 3_o(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))

Theorem

clsk1indlem4 45137*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Theorem

clsk1indlem1 45138*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∃𝑠 ∈ 𝒫 3_o∃𝑡 ∈ 𝒫 3_o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))

Theorem

clsk1independent 45139*

For generalized closure functions, property K1 (isotony) is independent of the properties K0, K2, K3, K4. This contradicts a claim which appears in preprints of Table 2 in Bärbel M. R. Stadler and Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci., 42:577-585, 2002. Proceedings MCC 2001, Dubrovnik. The same table row implying K1 follows from the other four appears in the supplemental materials Bärbel M. R. Stadler and Peter F. Stadler. "Basic Properties of Closure Spaces" 2001 on page 12. (Contributed by RP, 5-Jul-2021.)

⊢ (𝜑 ↔ (𝑘‘∅) = ∅)    &   ⊢ (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))    &   ⊢ (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠))    &   ⊢ (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)))    &   ⊢ (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))    ⇒   ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)

Theorem

neik0pk1imk0 45140*

Kuratowski's K0' and K1 axioms imply K0. Neighborhood version. (Contributed by RP, 3-Jun-2021.)

⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑_m 𝐵))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥))

Theorem

isotone1 45141*

Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)

⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))

Theorem

isotone2 45142*

Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)

⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)))

Theorem

ntrk1k3eqk13 45143*

An interior function is both monotone and sub-linear if and only if it is finitely linear. (Contributed by RP, 18-Jun-2021.)

⊢ ((∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡))) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)))

Theorem

ntrclsf1o 45144*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑_m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

ntrclsnvobr 45145*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then they are related the opposite way. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐾𝐷𝐼)

Theorem

ntrclsiex 45146*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

ntrclskex 45147*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

ntrclsfv1 45148*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)

Theorem

ntrclsfv2 45149*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼)

Theorem

ntrclselnel1 45150*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 28-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))

Theorem

ntrclselnel2 45151*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in interior of the complement of a set and non-membership in the closure of the set. (Contributed by RP, 28-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑆)))

Theorem

ntrclsfv 45152*

The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))

Theorem

ntrclsfveq1 45153*

If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶)))

Theorem

ntrclsfveq2 45154*

If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶)))

Theorem

ntrclsfveq 45155*

If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))

Theorem

ntrclsss 45156*

If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆))))

Theorem

ntrclsneine0lem 45157*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that at least one (pseudo-)neighborbood of a particular point exists hold equally. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠)))

Theorem

ntrclsneine0 45158*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 21-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐾‘𝑠)))

Theorem

ntrclscls00 45159*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Theorem

ntrclsiso 45160*

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡))))

Theorem

ntrclsk2 45161*

An interior function is contracting if and only if the closure function is expansive. (Contributed by RP, 9-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))

Theorem

ntrclskb 45162*

The interiors of disjoint sets are disjoint if and only if the closures of sets that span the base set also span the base set. (Contributed by RP, 10-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = 𝐵)))

Theorem

ntrclsk3 45163*

The intersection of interiors of a every pair is a subset of the interior of the intersection of the pair if an only if the closure of the union of every pair is a subset of the union of closures of the pair. (Contributed by RP, 19-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))

Theorem

ntrclsk13 45164*

The interior of the intersection of any pair is equal to the intersection of the interiors if and only if the closure of the unions of any pair is equal to the union of closures. (Contributed by RP, 19-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))

Theorem

ntrclsk4 45165*

Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-2021.)

⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑_m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))

Theorem

ntrneibex 45166*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the base set exists. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)

Theorem

ntrneircomplex 45167*

The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Theorem

ntrneif1o 45168*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑_m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑_m 𝐵))

Theorem

ntrneiiex 45169*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the interior function exists. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

ntrneinex 45170*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the neighborhood function exists. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑_m 𝐵))

Theorem

ntrneicnv 45171*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then converse of 𝐹 is known. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → ^◡𝐹 = (𝐵𝑂𝒫 𝐵))

Theorem

ntrneifv1 45172*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of 𝐹 is the neighborhood function. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)

Theorem

ntrneifv2 45173*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of converse of 𝐹 is the interior function. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (^◡𝐹‘𝑁) = 𝐼)

Theorem

ntrneiel 45174*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋)))

Theorem

ntrneifv3 45175*

The value of the neighbors (convergents) expressed in terms of the interior (closure) function. (Contributed by RP, 26-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)})

Theorem

ntrneineine0lem 45176*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ ∅))

Theorem

ntrneineine1lem 45177*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))

Theorem

ntrneifv4 45178*

The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)})

Theorem

ntrneiel2 45179*

Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (Contributed by RP, 11-Jul-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))

Theorem

ntrneineine0 45180*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅))

Theorem

ntrneineine1 45181*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ 𝒫 𝐵))

Theorem

ntrneicls00 45182*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))

Theorem

ntrneicls11 45183*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ ∅ ∈ (𝑁‘𝑥)))

Theorem

ntrneiiso 45184*

If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior function is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))

Theorem

ntrneik2 45185*

An interior function is contracting if and only if all the neighborhoods of a point contain that point. (Contributed by RP, 11-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠)))

Theorem

ntrneix2 45186*

An interior (closure) function is expansive if and only if all subsets which contain a point are neighborhoods (convergents) of that point. (Contributed by RP, 11-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑁‘𝑥))))

Theorem

ntrneikb 45187*

The interiors of disjoint sets are disjoint if and only if the neighborhoods of every point contain no disjoint sets. (Contributed by RP, 11-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ≠ ∅)))

Theorem

ntrneixb 45188*

The interiors (closures) of sets that span the base set also span the base set if and only if the neighborhoods (convergents) of every point contain at least one of every pair of sets that span the base set. (Contributed by RP, 11-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))

Theorem

ntrneik3 45189*

The intersection of interiors of any pair is a subset of the interior of the intersection if and only if the intersection of any two neighborhoods of a point is also a neighborhood. (Contributed by RP, 19-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥))))

Theorem

ntrneix3 45190*

The closure of the union of any pair is a subset of the union of closures if and only if the union of any pair belonging to the convergents of a point implies at least one of the pair belongs to the the convergents of that point. (Contributed by RP, 19-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))

Theorem

ntrneik13 45191*

The interior of the intersection of any pair equals intersection of interiors if and only if the intersection of any pair belonging to the neighborhood of a point is equivalent to both of the pair belonging to the neighborhood of that point. (Contributed by RP, 19-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)))))

Theorem

ntrneix13 45192*

The closure of the union of any pair is equal to the union of closures if and only if the union of any pair belonging to the convergents of a point if equivalent to at least one of the pain belonging to the convergents of that point. (Contributed by RP, 19-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))

Theorem

ntrneik4w 45193*

Idempotence of the interior function is equivalent to saying a set is a neighborhood of a point if and only if the interior of the set is a neighborhood of a point. (Contributed by RP, 11-Jul-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))

Theorem

ntrneik4 45194*

Idempotence of the interior function is equivalent to stating a set, 𝑠, is a neighborhood of a point, 𝑥 is equivalent to there existing a special neighborhood, 𝑢, of 𝑥 such that a point is an element of the special neighborhood if and only if 𝑠 is also a neighborhood of the point. (Contributed by RP, 11-Jul-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))))

Theorem

clsneibex 45195

If (pseudo-)closure and (pseudo-)neighborhood functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)

⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)

Theorem

clsneircomplex 45196

The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)

⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Theorem

clsneif1o 45197*

If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the operator is a one-to-one, onto mapping. (Contributed by RP, 5-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑_m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → 𝐻:(𝒫 𝐵 ↑_m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑_m 𝐵))

Theorem

clsneicnv 45198*

If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the converse of the operator is known. (Contributed by RP, 5-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑_m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → ^◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))

Theorem

clsneikex 45199*

If closure and neighborhoods functions are related, the closure function exists. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑_m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

clsneinex 45200*

If closure and neighborhoods functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.)

⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑_m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑_m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑_m 𝐵))