P. 242 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hmph 24101	Express the predicate 𝐽 is homeomorphic to 𝐾. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
Theorem	hmphi 24102	If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐽 ≃ 𝐾)
Theorem	hmphtop 24103	Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
Theorem	hmphtop1 24104	The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → 𝐽 ∈ Top)
Theorem	hmphtop2 24105	The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → 𝐾 ∈ Top)
Theorem	hmphref 24106	"Is homeomorphic to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐽 ∈ Top → 𝐽 ≃ 𝐽)
Theorem	hmphsym 24107	"Is homeomorphic to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)
⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽)
Theorem	hmphtr 24108	"Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐽 ≃ 𝐾 ∧ 𝐾 ≃ 𝐿) → 𝐽 ≃ 𝐿)
Theorem	hmpher 24109	"Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ ≃ Er Top
Theorem	hmphen 24110	Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐽 ≃ 𝐾 → 𝐽 ≈ 𝐾)
Theorem	hmphsymb 24111	"Is homeomorphic to" is symmetric. (Contributed by FL, 22-Feb-2007.)
⊢ (𝐽 ≃ 𝐾 ↔ 𝐾 ≃ 𝐽)
Theorem	haushmphlem 24112*	Lemma for haushmph 24117 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)    &   ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴)    ⇒   ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
Theorem	cmphmph 24113	Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
Theorem	connhmph 24114	Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.)
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Conn → 𝐾 ∈ Conn))
Theorem	t0hmph 24115	T0 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Kol2 → 𝐾 ∈ Kol2))
Theorem	t1hmph 24116	T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre))
Theorem	haushmph 24117	Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Haus → 𝐾 ∈ Haus))
Theorem	reghmph 24118	Regularity is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
Theorem	nrmhmph 24119	Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
Theorem	hmph0 24120	A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅})
Theorem	hmphdis 24121	Homeomorphisms preserve topological discreteness. (Contributed by Mario Carneiro, 10-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋)
Theorem	hmphindis 24122	Homeomorphisms preserve topological indiscreteness. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ≃ {∅, 𝐴} → 𝐽 = {∅, 𝑋})
Theorem	indishmph 24123	Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
⊢ (𝐴 ≈ 𝐵 → {∅, 𝐴} ≃ {∅, 𝐵})
Theorem	hmphen2 24124	Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌)
Theorem	cmphaushmeo 24125	A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹:𝑋–1-1-onto→𝑌))
Theorem	ordthmeolem 24126	Lemma for ordthmeo 24127. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ 𝑌 = dom 𝑆    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅) Cn (ordTop‘𝑆)))
Theorem	ordthmeo 24127	An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    &   ⊢ 𝑌 = dom 𝑆    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐹 Isom 𝑅, 𝑆 (𝑋, 𝑌)) → 𝐹 ∈ ((ordTop‘𝑅)Homeo(ordTop‘𝑆)))
Theorem	txhmeo 24128*	Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo𝐿))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀)))
Theorem	txswaphmeolem 24129*	Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
Theorem	txswaphmeo 24130*	There is a homeomorphism from 𝑋 × 𝑌 to 𝑌 × 𝑋. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))
Theorem	pt1hmeo 24131*	The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.) (Proof shortened by AV, 18-Apr-2021.)
⊢ 𝐾 = (∏t‘{⟨𝐴, 𝐽⟩})    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))
Theorem	ptuncnv 24132*	Exhibit the converse function of the map 𝐺 which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ 𝑋 = ∪ 𝐾    &   ⊢ 𝑌 = ∪ 𝐿    &   ⊢ 𝐽 = (∏t‘𝐹)    &   ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐴))    &   ⊢ 𝐿 = (∏t‘(𝐹 ↾ 𝐵))    &   ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐶⟶Top)    &   ⊢ (𝜑 → 𝐶 = (𝐴 ∪ 𝐵))    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → ◡𝐺 = (𝑧 ∈ ∪ 𝐽 ↦ ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩))
Theorem	ptunhmeo 24133*	Define a homeomorphism from a binary product of indexed product topologies to an indexed product topology on the union of the index sets. This is the topological analogue of (𝐴↑𝐵) · (𝐴↑𝐶) = 𝐴↑(𝐵 + 𝐶). (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ 𝑋 = ∪ 𝐾    &   ⊢ 𝑌 = ∪ 𝐿    &   ⊢ 𝐽 = (∏t‘𝐹)    &   ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐴))    &   ⊢ 𝐿 = (∏t‘(𝐹 ↾ 𝐵))    &   ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐶⟶Top)    &   ⊢ (𝜑 → 𝐶 = (𝐴 ∪ 𝐵))    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐾 ×t 𝐿)Homeo𝐽))
Theorem	xpstopnlem1 24134*	The function 𝐹 used in xpsval 17748 is a homeomorphism from the binary product topology to the indexed product topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))
Theorem	xpstps 24135	A binary product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    ⇒   ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 ∈ TopSp)
Theorem	xpstopnlem2 24136*	Lemma for xpstopn 24137. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝐾 = (TopOpen‘𝑆)    &   ⊢ 𝑂 = (TopOpen‘𝑇)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    ⇒   ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))
Theorem	xpstopn 24137	The topology on a binary product of topological spaces, as we have defined it (transferring the indexed product topology on functions on {∅, 1o} to (𝑋 × 𝑌) by the canonical bijection), coincides with the usual topological product (generated by a base of rectangles). (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝐾 = (TopOpen‘𝑆)    &   ⊢ 𝑂 = (TopOpen‘𝑇)    ⇒   ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))
Theorem	ptcmpfi 24138	A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘𝐹) ∈ Comp)
Theorem	xkocnv 24139*	The inverse of the "currying" function 𝐹 is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))    &   ⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐿 ∈ Top)    ⇒   ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
Theorem	xkohmeo 24140*	The Exponential Law for topological spaces. The "currying" function 𝐹 is a homeomorphism on function spaces when 𝐽 and 𝐾 are exponentiable spaces (by xkococn 23985, it is sufficient to assume that 𝐽, 𝐾 are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))    &   ⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐿 ∈ Top)    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ↑ko (𝐽 ×t 𝐾))Homeo((𝐿 ↑ko 𝐾) ↑ko 𝐽)))
Theorem	qtopf1 24141	If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
Theorem	qtophmeo 24142*	If two functions on a base topology 𝐽 make the same identifications in order to create quotient spaces 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺, then not only are 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺 homeomorphic, but there is a unique homeomorphism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺:𝑋–onto→𝑌)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐺‘𝑥) = (𝐺‘𝑦)))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
Theorem	t0kq 24143*	A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))
Theorem	kqhmph 24144	A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
Theorem	ist1-5lem 24145	Lemma for ist1-5 24147 and similar theorems. If 𝐴 is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property 𝐴 (which is defined as stating that the Kolmogorov quotient of the space has property 𝐴). For example, if 𝐴 is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Kol2)    &   ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ 𝐴 → (KQ‘𝐽) ∈ 𝐴))    &   ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴 → 𝐽 ∈ 𝐴))    ⇒   ⊢ (𝐽 ∈ 𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
Theorem	t1r0 24146	A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre)
Theorem	ist1-5 24147	A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Fre))
Theorem	ishaus3 24148	A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus))
Theorem	nrmreg 24149	A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 24074. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg)
Theorem	reghaus 24150	A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ (𝐽 ∈ Reg → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))
Theorem	nrmhaus 24151	A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
12.2  Filters and filter bases
12.2.1  Filter bases
Theorem	elmptrab 24152*	Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})    &   ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    &   ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓))
Theorem	elmptrab2 24153*	Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Revised by AV, 26-Mar-2021.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})    &   ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓))
Theorem	isfbas 24154*	The predicate "𝐹 is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))
Theorem	fbasne0 24155	There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ≠ ∅)
Theorem	0nelfb 24156	No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹)
Theorem	fbsspw 24157	A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ⊆ 𝒫 𝐵)
Theorem	fbelss 24158	An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑋 ∈ 𝐹) → 𝑋 ⊆ 𝐵)
Theorem	fbdmn0 24159	The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐵 ≠ ∅)
Theorem	isfbas2 24160*	The predicate "𝐹 is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
Theorem	fbasssin 24161*	A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))
Theorem	fbssfi 24162*	A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)
Theorem	fbssint 24163*	A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴)
Theorem	fbncp 24164	A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝐵 ∖ 𝐴) ∈ 𝐹)
Theorem	fbun 24165*	A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
Theorem	fbfinnfr 24166	No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin) → ∩ 𝐹 ≠ ∅)
Theorem	opnfbas 24167*	The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋))
Theorem	trfbas2 24168	Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹 ↾t 𝐴)))
Theorem	trfbas 24169*	Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ∀𝑣 ∈ 𝐹 (𝑣 ∩ 𝐴) ≠ ∅))
12.2.2  Filters
Syntax	cfil 24170	Extend class notation with the set of filters on a set.
class Fil
Definition	df-fil 24171*	The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in ℝ. With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
Theorem	isfil 24172*	The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹)))
Theorem	filfbas 24173	A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
Theorem	0nelfil 24174	The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
Theorem	fileln0 24175	An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ≠ ∅)
Theorem	filsspw 24176	A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
Theorem	filelss 24177	An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
Theorem	filss 24178	A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐵 ∈ 𝐹)
Theorem	filin 24179	A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ∩ 𝐵) ∈ 𝐹)
Theorem	filtop 24180	The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
Theorem	isfil2 24181*	Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) ↔ ((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹))
Theorem	isfildlem 24182*	Lemma for isfild 24183. (Contributed by Mario Carneiro, 1-Dec-2013.)
⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))
Theorem	isfild 24183*	Sufficient condition for a set of the form {𝑥 ∈ 𝒫 𝐴 ∣ 𝜑} to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) (Revised by AV, 10-Apr-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)    &   ⊢ (𝜑 → ¬ [∅ / 𝑥]𝜓)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜓))    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (Fil‘𝐴))
Theorem	filfi 24184	A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
Theorem	filinn0 24185	The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ∩ 𝐵) ≠ ∅)
Theorem	filintn0 24186	A filter has the finite intersection property. Remark below Definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ⊆ 𝐹 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ≠ ∅)
Theorem	filn0 24187	The empty set is not a filter. Remark below Definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
Theorem	infil 24188	The intersection of two filters is a filter. Use fiint 9322 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ∈ (Fil‘𝑋))
Theorem	snfil 24189	A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))
Theorem	fbasweak 24190	A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑌))
Theorem	snfbas 24191	Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ 𝑉) → {𝐴} ∈ (fBas‘𝐵))
Theorem	fsubbas 24192	A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝑋 ∈ 𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))
Theorem	fbasfip 24193	A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
Theorem	fbunfip 24194*	A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
Theorem	fgval 24195*	The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})
Theorem	elfg 24196*	A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐴 ∈ (𝑋filGen𝐹) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)))
Theorem	ssfg 24197	A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
Theorem	fgss 24198	A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
Theorem	fgss2 24199*	A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
Theorem	fgfil 24200	A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)