P. 244 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	supnfcls 24301*	The filter of supersets of 𝑋 ∖ 𝑈 does not cluster at any point of the open set 𝑈. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑈) ⊆ 𝑥}))
Theorem	fclsbas 24302*	Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ 𝐹 = (𝑋filGen𝐵)    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))))
Theorem	fclsss1 24303	A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))
Theorem	fclsss2 24304	A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))
Theorem	fclsrest 24305	The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾t 𝑌) fClus (𝐹 ↾t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌))
Theorem	fclscf 24306*	Characterization of fineness of topologies in terms of cluster points. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)))
Theorem	flimfcls 24307	A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
Theorem	fclsfnflim 24308*	A filter clusters at a point iff a finer filter converges to it. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
Theorem	flimfnfcls 24309*	A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 24308, this theorem illustrates the duality between convergence and clustering. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))))
Theorem	fclscmpi 24310	Forward direction of fclscmp 24311. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)
Theorem	fclscmp 24311*	A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
Theorem	uffclsflim 24312	The cluster points of an ultrafilter are its limit points. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))
Theorem	ufilcmp 24313*	A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
Theorem	fcfval 24314	The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
Theorem	isfcf 24315*	The property of being a cluster point of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))))
Theorem	fcfnei 24316*	The property of being a cluster point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
Theorem	fcfelbas 24317	A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹)) → 𝐴 ∈ 𝑋)
Theorem	fcfneii 24318	A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿)) → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)
Theorem	flfssfcf 24319	A limit point of a function is a cluster point of the function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) ⊆ ((𝐽 fClusf 𝐿)‘𝐹))
Theorem	uffcfflf 24320	If the domain filter is an ultrafilter, the cluster points of the function are the limit points. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = ((𝐽 fLimf 𝐿)‘𝐹))
Theorem	cnpfcfi 24321	Lemma for cnpfcf 24322. If a function is continuous at a point, it respects clustering there. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))
Theorem	cnpfcf 24322*	A function 𝐹 is continuous at point 𝐴 iff 𝐹 respects cluster points there. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
Theorem	cnfcf 24323*	Continuity of a function in terms of cluster points of a function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
Theorem	flfcntr 24324	A continuous function's value is always in the trace of its filter limit. (Contributed by Thierry Arnoux, 30-Aug-2020.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
Theorem	alexsublem 24325*	Lemma for alexsub 24326. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝜑 → 𝑋 ∈ UFL)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐵)    &   ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵)))    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)    &   ⊢ (𝜑 → 𝐹 ∈ (UFil‘𝑋))    &   ⊢ (𝜑 → (𝐽 fLim 𝐹) = ∅)    ⇒   ⊢ ¬ 𝜑
Theorem	alexsub 24326*	The Alexander Subbase Theorem: If 𝐵 is a subbase for the topology 𝐽, and any cover taken from 𝐵 has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 24332 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ (𝜑 → 𝑋 ∈ UFL)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐵)    &   ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵)))    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)    ⇒   ⊢ (𝜑 → 𝐽 ∈ Comp)
Theorem	alexsubb 24327*	Biconditional form of the Alexander Subbase Theorem alexsub 24326. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp ↔ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))
Theorem	alexsubALTlem1 24328*	Lemma for alexsubALT 24332. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
Theorem	alexsubALTlem2 24329*	Lemma for alexsubALT 24332. Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff Hankins, 27-Jan-2010.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) → ∃𝑢 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ¬ 𝑢 ⊊ 𝑣)
Theorem	alexsubALTlem3 24330*	Lemma for alexsubALT 24332. If a point is covered by a collection taken from the base with no finite subcover, a set from the subbase can be added that covers the point so that the resulting collection has no finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑢 ∧ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) ∧ 𝑤 ∈ 𝑢) ∧ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢)))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)
Theorem	alexsubALTlem4 24331*	Lemma for alexsubALT 24332. If any cover taken from a subbase has a finite subcover, any cover taken from the corresponding base has a finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) → ∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
Theorem	alexsubALT 24332*	The Alexander Subbase Theorem: a space is compact iff it has a subbase such that any cover taken from the subbase has a finite subcover. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Comp ↔ ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
Theorem	ptcmplem1 24333*	Lemma for ptcmp 24339. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))    &   ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Comp)    &   ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))    ⇒   ⊢ (𝜑 → (𝑋 = ∪ (ran 𝑆 ∪ {𝑋}) ∧ (∏t‘𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
Theorem	ptcmplem2 24334*	Lemma for ptcmp 24339. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))    &   ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Comp)    &   ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))    &   ⊢ (𝜑 → 𝑈 ⊆ ran 𝑆)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)
Theorem	ptcmplem3 24335*	Lemma for ptcmp 24339. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))    &   ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Comp)    &   ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))    &   ⊢ (𝜑 → 𝑈 ⊆ ran 𝑆)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)    &   ⊢ 𝐾 = {𝑢 ∈ (𝐹‘𝑘) ∣ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑈}    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
Theorem	ptcmplem4 24336*	Lemma for ptcmp 24339. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))    &   ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Comp)    &   ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))    &   ⊢ (𝜑 → 𝑈 ⊆ ran 𝑆)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)    &   ⊢ 𝐾 = {𝑢 ∈ (𝐹‘𝑘) ∣ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑈}    ⇒   ⊢ ¬ 𝜑
Theorem	ptcmplem5 24337*	Lemma for ptcmp 24339. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))    &   ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Comp)    &   ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))    ⇒   ⊢ (𝜑 → (∏t‘𝐹) ∈ Comp)
Theorem	ptcmpg 24338	Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if 𝒫 𝒫 𝑋 is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 24339). (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ 𝐽 = (∏t‘𝐹)    &   ⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)
Theorem	ptcmp 24339	Tychonoff's theorem: The product of compact spaces is compact. The proof uses the Axiom of Choice. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp) → (∏t‘𝐹) ∈ Comp)
12.2.5  Extension by continuity
Syntax	ccnext 24340	Extend class notation with the continuous extension operation.
class CnExt
Definition	df-cnext 24341*	Define the continuous extension of a given function. (Contributed by Thierry Arnoux, 1-Dec-2017.)
⊢ CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (∪ 𝑘 ↑pm ∪ 𝑗) ↦ ∪ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
Theorem	cnextval 24342*	The function applying continuous extension to a given function 𝑓. (Contributed by Thierry Arnoux, 1-Dec-2017.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
Theorem	cnextfval 24343*	The continuous extension of a given function 𝐹. (Contributed by Thierry Arnoux, 1-Dec-2017.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
Theorem	cnextrel 24344	In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹))
Theorem	cnextfun 24345	If the target space is Hausdorff, a continuous extension is a function. (Contributed by Thierry Arnoux, 20-Dec-2017.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    ⇒   ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
Theorem	cnextfvval 24346*	The value of the continuous extension of a given function 𝐹 at a point 𝑋. (Contributed by Thierry Arnoux, 21-Dec-2017.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
Theorem	cnextf 24347*	Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)    ⇒   ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)
Theorem	cnextcn 24348*	Extension by continuity. Theorem 1 of [BourbakiTop1] p. I.57. Given a topology 𝐽 on 𝐶, a subset 𝐴 dense in 𝐶, this states a condition for 𝐹 from 𝐴 to a regular space 𝐾 to be extensible by continuity. (Contributed by Thierry Arnoux, 1-Jan-2018.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)    &   ⊢ (𝜑 → 𝐾 ∈ Reg)    ⇒   ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
Theorem	cnextfres1 24349*	𝐹 and its extension by continuity agree on the domain of 𝐹. (Contributed by Thierry Arnoux, 17-Jan-2018.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)    &   ⊢ (𝜑 → 𝐾 ∈ Reg)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))    ⇒   ⊢ (𝜑 → (((𝐽CnExt𝐾)‘𝐹) ↾ 𝐴) = 𝐹)
Theorem	cnextfres 24350	𝐹 and its extension by continuity agree on the domain of 𝐹. (Contributed by Thierry Arnoux, 29-Aug-2020.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = (𝐹‘𝑋))
12.2.6  Topological groups
Syntax	ctmd 24351	Extend class notation with the class of all topological monoids.
class TopMnd
Syntax	ctgp 24352	Extend class notation with the class of all topological groups.
class TopGrp
Definition	df-tmd 24353*	Define the class of all topological monoids. A topological monoid is a monoid whose operation is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
Definition	df-tgp 24354*	Define the class of all topological groups. A topological group is a group whose operation and inverse function are continuous. (Contributed by FL, 18-Apr-2010.)
⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)}
Theorem	istmd 24355	The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐹 = (+𝑓‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
Theorem	tmdmnd 24356	A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
Theorem	tmdtps 24357	A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
Theorem	istgp 24358	The predicate "is a topological group". Definition 1 of [BourbakiTop1] p. III.1. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
Theorem	tgpgrp 24359	A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
Theorem	tgptmd 24360	A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
Theorem	tgptps 24361	A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
Theorem	tmdtopon 24362	The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
Theorem	tgptopon 24363	The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
Theorem	tmdcn 24364	In a topological monoid, the operation 𝐹 representing the functionalization of the operator slot +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐹 = (+𝑓‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Theorem	tgpcn 24365	In a topological group, the operation 𝐹 representing the functionalization of the operator slot +g is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐹 = (+𝑓‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Theorem	tgpinv 24366	In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
Theorem	grpinvhmeo 24367	The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽))
Theorem	cnmpt1plusg 24368*	Continuity of the group sum; analogue of cnmpt12f 23947 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
Theorem	cnmpt2plusg 24369*	Continuity of the group sum; analogue of cnmpt22f 23956 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Theorem	tmdcn2 24370*	Write out the definition of continuity of +g explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))
Theorem	tgpsubcn 24371	In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Theorem	istgp2 24372	A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
Theorem	tmdmulg 24373*	In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopMnd ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽))
Theorem	tgpmulg 24374*	In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽))
Theorem	tgpmulg2 24375	In a topological monoid, the group multiple function is jointly continuous (although this is not saying much as one of the factors is discrete). Use zdis 25098 to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → · ∈ ((𝒫 ℤ ×t 𝐽) Cn 𝐽))
Theorem	tmdgsum 24376*	In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when 𝐴 is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵 ↑m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ↑ko 𝒫 𝐴) Cn 𝐽))
Theorem	tmdgsum2 24377*	For any neighborhood 𝑈 of 𝑛𝑋, there is a neighborhood 𝑢 of 𝑋 such that any sum of 𝑛 elements in 𝑢 sums to an element of 𝑈. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑m 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
Theorem	oppgtmd 24378	The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd)
Theorem	oppgtgp 24379	The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → 𝑂 ∈ TopGrp)
Theorem	distgp 24380	Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)
Theorem	indistgp 24381	Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp)
Theorem	efmndtmd 24382	The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 24387. (Contributed by AV, 23-Feb-2024.)
⊢ 𝑀 = (EndoFMnd‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd)
Theorem	tmdlactcn 24383*	The left group action of element 𝐴 in a topological monoid 𝐺 is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
Theorem	tgplacthmeo 24384*	The left group action of element 𝐴 in a topological group 𝐺 is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))
Theorem	submtmd 24385	A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)
Theorem	subgtgp 24386	A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)
Theorem	symgtgp 24387	The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.) (Proof shortened by AV, 30-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopGrp)
Theorem	subgntr 24388	A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 24390, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐽)
Theorem	opnsubg 24389	An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))
Theorem	clssubg 24390	The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))
Theorem	clsnsg 24391	The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))
Theorem	cldsubg 24392	A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑅 = (𝐺 ~QG 𝑆)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ 𝐽))
Theorem	tgpconncompeqg 24393*	The connected component containing 𝐴 is the left coset of the identity component containing 𝐴. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
Theorem	tgpconncomp 24394*	The identity component, the connected component containing the identity element, is a closed (conncompcld 23714) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}    ⇒   ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))
Theorem	tgpconncompss 24395*	The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝑆 ⊆ 𝑇)
Theorem	ghmcnp 24396	A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐾 = (TopOpen‘𝐻)    ⇒   ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
Theorem	snclseqg 24397	The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    &   ⊢ 𝑆 = ((cls‘𝐽)‘{ 0 })    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴}))
Theorem	tgphaus 24398	A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽)))
Theorem	tgpt1 24399	Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
Theorem	tgpt0 24400	Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))