P. 339 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33801-33900   *Has distinct variable group(s)

Type	Label	Description
Statement
21.3.14.2  Topology of the unit circle
Theorem	qtopt1 33801*	If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre)
Theorem	qtophaus 33802*	If an open map's graph in the product space (𝐽 ×t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ ∼ = (◡𝐹 ∘ 𝐹)    &   ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ Haus)    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹))    &   ⊢ (𝜑 → ∼ ∈ (Clsd‘(𝐽 ×t 𝐽)))    ⇒   ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Haus)
Theorem	circtopn 33803*	The topology of the unit circle is generated by open intervals of the polar coordinate. (Contributed by Thierry Arnoux, 4-Jan-2020.)
⊢ 𝐼 = (0[,](2 · π))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   ⊢ 𝐶 = (◡abs “ {1})    ⇒   ⊢ (𝐽 qTop 𝐹) = (TopOpen‘(𝐹 “s ℝfld))
Theorem	circcn 33804*	The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-Jan-2020.)
⊢ 𝐼 = (0[,](2 · π))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   ⊢ 𝐶 = (◡abs “ {1})    ⇒   ⊢ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))
21.3.14.3  Refinements
Theorem	reff 33805*	For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a definition of refinement. Note that this definition uses the axiom of choice through ac6sg 10401. (Contributed by Thierry Arnoux, 12-Jan-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))
Theorem	locfinreflem 33806*	A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑉Ref𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
Theorem	locfinref 33807*	A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑉Ref𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
21.3.14.4  Open cover refinement property
Syntax	ccref 33808	The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴
Definition	df-cref 33809*	Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
Theorem	iscref 33810*	The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
Theorem	crefeq 33811	Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)
Theorem	creftop 33812	A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top)
Theorem	crefi 33813*	The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
Theorem	crefdf 33814*	A formulation of crefi 33813 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐵 = CovHasRef𝐴    &   ⊢ (𝑧 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))
Theorem	crefss 33815	The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)
Theorem	cmpcref 33816	Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ Comp = CovHasRefFin
Theorem	cmpfiref 33817*	Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈))
21.3.14.5  Lindelöf spaces
Syntax	cldlf 33818	Extend class notation with the class of all Lindelöf spaces.
class Ldlf
Definition	df-ldlf 33819	Definition of a Lindelöf space. A Lindelöf space is a topological space in which every open cover has a countable subcover. Definition 1 of [BourbakiTop2] p. 195. (Contributed by Thierry Arnoux, 30-Jan-2020.)
⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
Theorem	ldlfcntref 33820*	Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
21.3.14.6  Paracompact spaces
Syntax	cpcmp 33821	Extend class notation with the class of all paracompact topologies.
class Paracomp
Definition	df-pcmp 33822	Definition of a paracompact topology. A topology is said to be paracompact iff every open cover has an open refinement that is locally finite. The definition 6 of [BourbakiTop1] p. I.69. also requires the topology to be Hausdorff, but this is dropped here. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
Theorem	ispcmp 33823	The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
Theorem	cmppcmp 33824	Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)
Theorem	dispcmp 33825	Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp)
Theorem	pcmplfin 33826*	Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
Theorem	pcmplfinf 33827*	Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover 𝑈. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
21.3.14.7  Spectrum of a ring The prime ideals of a ring 𝑅 can be endowed with the Zariski topology. This is done by defining a function 𝑉 which maps ideals of 𝑅 to closed sets (see for example zarcls0 33834 for the definition of 𝑉). The closed sets of the topology are in the range of 𝑉 (see zartopon 33843). The correspondence with the open sets is made in zarcls 33840. As proved in zart0 33845, the Zariski topology is T0 , but generally not T1 .
Syntax	crspec 33828	Extend class notation with the spectrum of a ring.
class Spec
Definition	df-rspec 33829	Define the spectrum of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
Theorem	rspecval 33830	Value of the spectrum of the ring 𝑅. Notation 1.1.1 of [EGA] p. 80. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
Theorem	rspecbas 33831	The prime ideals form the base of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = (Base‘𝑆))
Theorem	rspectset 33832*	Topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = (TopSet‘𝑆))
Theorem	rspectopn 33833*	The topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 4-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))
Theorem	zarcls0 33834*	The closure of the identity ideal in the Zariski topology. Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑉‘{ 0 }) = 𝑃)
Theorem	zarcls1 33835*	The unit ideal 𝐵 is the only ideal whose closure in the Zariski topology is the empty set. Stronger form of the Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝐼) = ∅ ↔ 𝐼 = 𝐵))
Theorem	zarclsun 33836*	The union of two closed sets of the Zariski topology is closed. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
Theorem	zarclsiin 33837*	In a Zariski topology, the intersection of the closures of a family of ideals is the closure of the span of their union. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐾 = (RSpan‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) = (𝑉‘(𝐾‘∪ 𝑇)))
Theorem	zarclsint 33838*	The intersection of a family of closed sets is closed in the Zariski topology. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)
Theorem	zarclssn 33839*	The closed points of Zariski topology are the maximal ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐵 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
Theorem	zarcls 33840*	The open sets of the Zariski topology are the complements of the closed sets. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
Theorem	zartopn 33841*	The Zariski topology is a topology, and its closed sets are images by 𝑉 of the ideals of 𝑅. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)))
Theorem	zartop 33842	The Zariski topology is a topology. Proposition 1.1.2 of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
Theorem	zartopon 33843	The points of the Zariski topology are the prime ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝑃))
Theorem	zar0ring 33844	The Zariski Topology of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐽 = {∅})
Theorem	zart0 33845	The Zariski topology is T0 . Corollary 1.1.8 of [EGA] p. 81. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2)
Theorem	zarmxt1 33846	The Zariski topology restricted to maximal ideals is T1 . (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑀 = (MaxIdeal‘𝑅)    &   ⊢ 𝑇 = (𝐽 ↾t 𝑀)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)
Theorem	zarcmplem 33847*	Lemma for zarcmp 33848. (Contributed by Thierry Arnoux, 2-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)
Theorem	zarcmp 33848	The Zariski topology is compact. Proposition 1.1.10(ii) of [EGA], p. 82. (Contributed by Thierry Arnoux, 2-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)
Theorem	rspectps 33849	The spectrum of a ring 𝑅 is a topological space. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑆 ∈ TopSp)
Theorem	rhmpreimacnlem 33850*	Lemma for rhmpreimacn 33851. (Contributed by Thierry Arnoux, 7-Jul-2024.)
⊢ 𝑇 = (Spec‘𝑅)    &   ⊢ 𝑈 = (Spec‘𝑆)    &   ⊢ 𝐴 = (PrmIdeal‘𝑅)    &   ⊢ 𝐵 = (PrmIdeal‘𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑇)    &   ⊢ 𝐾 = (TopOpen‘𝑈)    &   ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆))    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ 𝑉 = (𝑗 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘})    &   ⊢ 𝑊 = (𝑗 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘})    ⇒   ⊢ (𝜑 → (𝑊‘(𝐹 “ 𝐼)) = (◡𝐺 “ (𝑉‘𝐼)))
Theorem	rhmpreimacn 33851*	The function mapping a prime ideal to its preimage by a surjective ring homomorphism is continuous, when considering the Zariski topology. Corollary 1.2.3 of [EGA], p. 83. Notice that the direction of the continuous map 𝐺 is reverse: the original ring homomorphism 𝐹 goes from 𝑅 to 𝑆, but the continuous map 𝐺 goes from 𝐵 to 𝐴. This mapping is also called "induced map on prime spectra" or "pullback on primes". (Contributed by Thierry Arnoux, 8-Jul-2024.)
⊢ 𝑇 = (Spec‘𝑅)    &   ⊢ 𝑈 = (Spec‘𝑆)    &   ⊢ 𝐴 = (PrmIdeal‘𝑅)    &   ⊢ 𝐵 = (PrmIdeal‘𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑇)    &   ⊢ 𝐾 = (TopOpen‘𝑈)    &   ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
21.3.14.8  Pseudometrics
Syntax	cmetid 33852	Extend class notation with the class of metric identifications.
class ~Met
Syntax	cpstm 33853	Extend class notation with the metric induced by a pseudometric.
class pstoMet
Definition	df-metid 33854*	Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ~Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
Definition	df-pstm 33855*	Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
Theorem	metidval 33856*	Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)})
Theorem	metidss 33857	As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋))
Theorem	metidv 33858	𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
Theorem	metideq 33859	Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))
Theorem	metider 33860	The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋)
Theorem	pstmval 33861*	Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ∼ = (~Met‘𝐷)    ⇒   ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}))
Theorem	pstmfval 33862	Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ∼ = (~Met‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = (𝐴𝐷𝐵))
Theorem	pstmxmet 33863	The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ∼ = (~Met‘𝐷)    ⇒   ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ )))
21.3.14.9  Continuity - misc additions
Theorem	hauseqcn 33864	In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑋)    &   ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
21.3.14.10  Topology of the closed unit interval
Theorem	elunitge0 33865	An element of the closed unit interval is positive. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 20-Dec-2016.)
⊢ (𝐴 ∈ (0[,]1) → 0 ≤ 𝐴)
Theorem	unitssxrge0 33866	The closed unit interval is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.)
⊢ (0[,]1) ⊆ (0[,]+∞)
Theorem	unitdivcld 33867	Necessary conditions for a quotient to be in the closed unit interval. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)
⊢ ((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1) ∧ 𝐵 ≠ 0) → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐵) ∈ (0[,]1)))
Theorem	iistmd 33868	The closed unit interval forms a topological monoid under multiplication. (Contributed by Thierry Arnoux, 25-Mar-2017.)
⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1))    ⇒   ⊢ 𝐼 ∈ TopMnd
21.3.14.11  Topology of ` ( RR X. RR ) `
Theorem	unicls 33869	The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ∪ (Clsd‘𝐽) = 𝑋
Theorem	tpr2tp 33870	The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ))
Theorem	tpr2uni 33871	The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ ∪ (𝐽 ×t 𝐽) = (ℝ × ℝ)
Theorem	xpinpreima 33872	Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵))
Theorem	xpinpreima2 33873	Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))
Theorem	sqsscirc1 33874	The complex square of side 𝐷 is a subset of the complex circle of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.)
⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ+) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷))
Theorem	sqsscirc2 33875	The complex square of side 𝐷 is a subset of the complex disc of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐷 ∈ ℝ+) → (((abs‘(ℜ‘(𝐵 − 𝐴))) < (𝐷 / 2) ∧ (abs‘(ℑ‘(𝐵 − 𝐴))) < (𝐷 / 2)) → (abs‘(𝐵 − 𝐴)) < 𝐷))
Theorem	cnre2csqlem 33876*	Lemma for cnre2csqima 33877. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ (𝐺 ↾ (ℝ × ℝ)) = (𝐻 ∘ 𝐹)    &   ⊢ 𝐹 Fn (ℝ × ℝ)    &   ⊢ 𝐺 Fn V    &   ⊢ (𝑥 ∈ (ℝ × ℝ) → (𝐺‘𝑥) ∈ ℝ)    &   ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹) → (𝐻‘(𝑥 − 𝑦)) = ((𝐻‘𝑥) − (𝐻‘𝑦)))    ⇒   ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ (◡(𝐺 ↾ (ℝ × ℝ)) “ (((𝐺‘𝑋) − 𝐷)(,)((𝐺‘𝑋) + 𝐷))) → (abs‘(𝐻‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷))
Theorem	cnre2csqima 33877*	Image of a centered square by the canonical bijection from (ℝ × ℝ) to ℂ. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    ⇒   ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)))
Theorem	tpr2rico 33878*	For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))    &   ⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))
21.3.14.12  Order topology - misc. additions
Theorem	cnvordtrestixx 33879*	The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.)
⊢ 𝐴 ⊆ ℝ*    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴)    ⇒   ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))
Theorem	prsdm 33880	Domain of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
Theorem	prsrn 33881	Range of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵)
Theorem	prsss 33882	Relation of a subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
Theorem	prsssdm 33883	Domain of a subproset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
Theorem	ordtprsval 33884*	Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})    &   ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})    ⇒   ⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸 ∪ 𝐹)))))
Theorem	ordtprsuni 33885*	Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})    &   ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})    ⇒   ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹)))
Theorem	ordtcnvNEW 33886	The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ))
Theorem	ordtrestNEW 33887	The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
Theorem	ordtrest2NEWlem 33888*	Lemma for ordtrest2NEW 33889. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
Theorem	ordtrest2NEW 33889*	An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴))
Theorem	ordtconnlem1 33890*	Connectedness in the order topology of a toset. This is the "easy" direction of ordtconn 33891. See also reconnlem1 24731. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ 𝐽 = (ordTop‘ ≤ )    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵) → ((𝐽 ↾t 𝐴) ∈ Conn → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑟 ∈ 𝐵 ((𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦) → 𝑟 ∈ 𝐴)))
Theorem	ordtconn 33891	Connectedness in the order topology of a complete uniform totally ordered space. (Contributed by Thierry Arnoux, 15-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ 𝐽 = (ordTop‘ ≤ )    ⇒   ⊢ ⊤
21.3.14.13  Continuity in topological spaces - misc. additions
Theorem	mndpluscn 33892*	A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.)
⊢ 𝐹 ∈ (𝐽Homeo𝐾)    &   ⊢ + :(𝐵 × 𝐵)⟶𝐵    &   ⊢ ∗ :(𝐶 × 𝐶)⟶𝐶    &   ⊢ 𝐽 ∈ (TopOn‘𝐵)    &   ⊢ 𝐾 ∈ (TopOn‘𝐶)    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))    &   ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)    ⇒   ⊢ ∗ ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
Theorem	mhmhmeotmd 33893	Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
⊢ 𝐹 ∈ (𝑆 MndHom 𝑇)    &   ⊢ 𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇))    &   ⊢ 𝑆 ∈ TopMnd    &   ⊢ 𝑇 ∈ TopSp    ⇒   ⊢ 𝑇 ∈ TopMnd
Theorem	rmulccn 33894*	Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) Avoid ax-mulf 11108. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽))
Theorem	raddcn 33895*	Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	xrmulc1cn 33896*	The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
⊢ 𝐽 = (ordTop‘ ≤ )    &   ⊢ 𝐹 = (𝑥 ∈ ℝ* ↦ (𝑥 ·e 𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐽))
Theorem	fmcncfil 33897	The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = (MetOpen‘𝐸)    ⇒   ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐸 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝐵 ∈ (CauFil‘𝐷)) → ((𝑌 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐸))
21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid
Theorem	xrge0hmph 33898	The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
⊢ II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞))
Theorem	xrge0iifcnv 33899*	Define a bijection from [0, 1] onto [0, +∞]. (Contributed by Thierry Arnoux, 29-Mar-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    ⇒   ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
Theorem	xrge0iifcv 33900*	The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    ⇒   ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋))