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Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfclselbas 21801 A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
𝑋 = 𝐽       (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴𝑋)

Theoremfclsneii 21802 A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁𝑆) ≠ ∅)

Theoremfclssscls 21803 The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝑆𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Theoremfclsnei 21804* Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅)))

Theoremsupnfcls 21805* The filter of supersets of 𝑋𝑈 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) ∧ 𝑈𝐽𝐴𝑈) → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}))

Theoremfclsbas 21806* 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) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐵 (𝑜𝑠) ≠ ∅))))

Theoremfclsss1 21807 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) ∧ 𝐽𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))

Theoremfclsss2 21808 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) ∧ 𝐹𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))

Theoremfclsrest 21809 The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐽t 𝑌) fClus (𝐹t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌))

Theoremfclscf 21810* 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)) → (𝐽𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)))

Theoremflimfcls 21811 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)

Theoremfclsfnflim 21812* 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)(𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔))))

Theoremflimfnfcls 21813* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 21812, 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)(𝐹𝑔𝐴 ∈ (𝐽 fClus 𝑔))))

Theoremfclscmpi 21814 Forward direction of fclscmp 21815. 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) ≠ ∅)

Theoremfclscmp 21815* 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) ≠ ∅))

Theoremuffclsflim 21816 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))

Theoremufilcmp 21817* 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) ≠ ∅))

Theoremfcfval 21818 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:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))

Theoremisfcf 21819* 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:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))))

Theoremfcfnei 21820* 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:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))

Theoremfcfelbas 21821 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:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹)) → 𝐴𝑋)

Theoremfcfneii 21822 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:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿)) → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)

Theoremflfssfcf 21823 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:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) ⊆ ((𝐽 fClusf 𝐿)‘𝐹))

Theoremuffcfflf 21824 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:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = ((𝐽 fLimf 𝐿)‘𝐹))

Theoremcnpfcfi 21825 Lemma for cnpfcf 21826. 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)) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))

Theoremcnpfcf 21826* 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) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))

Theoremcnfcf 21827* 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:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))

Theoremflfcntr 21828 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    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐴𝐶)    &   (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))    &   (𝜑𝑋𝐴)       (𝜑 → (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Theoremalexsublem 21829* Lemma for alexsub 21830. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝜑𝑋 ∈ UFL)    &   (𝜑𝑋 = 𝐵)    &   (𝜑𝐽 = (topGen‘(fi‘𝐵)))    &   ((𝜑 ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)    &   (𝜑𝐹 ∈ (UFil‘𝑋))    &   (𝜑 → (𝐽 fLim 𝐹) = ∅)        ¬ 𝜑

Theoremalexsub 21830* 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 21836 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
(𝜑𝑋 ∈ UFL)    &   (𝜑𝑋 = 𝐵)    &   (𝜑𝐽 = (topGen‘(fi‘𝐵)))    &   ((𝜑 ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)       (𝜑𝐽 ∈ Comp)

Theoremalexsubb 21831* Biconditional form of the Alexander Subbase Theorem alexsub 21830. (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)))

TheoremalexsubALTlem1 21832* Lemma for alexsubALT 21836. 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)))

TheoremalexsubALTlem2 21833* Lemma for alexsubALT 21836. 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) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ¬ 𝑢𝑣)

TheoremalexsubALTlem3 21834* Lemma for alexsubALT 21836. 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) ∧ (𝑎𝑢 ∧ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = 𝑏))) ∧ 𝑤𝑢) ∧ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = 𝑡) ∧ (𝑦𝑤 ∧ ¬ 𝑦 (𝑥𝑢)))) → ∃𝑠𝑡𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = 𝑛)

TheoremalexsubALTlem4 21835* Lemma for alexsubALT 21836. 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)))

TheoremalexsubALT 21836* 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)))

Theoremptcmplem1 21837* Lemma for ptcmp 21843. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))       (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))

Theoremptcmplem2 21838* Lemma for ptcmp 21843. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))    &   (𝜑𝑈 ⊆ ran 𝑆)    &   (𝜑𝑋 = 𝑈)    &   (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)       (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)

Theoremptcmplem3 21839* Lemma for ptcmp 21843. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))    &   (𝜑𝑈 ⊆ ran 𝑆)    &   (𝜑𝑋 = 𝑈)    &   (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)    &   𝐾 = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈}       (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))

Theoremptcmplem4 21840* Lemma for ptcmp 21843. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))    &   (𝜑𝑈 ⊆ ran 𝑆)    &   (𝜑𝑋 = 𝑈)    &   (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)    &   𝐾 = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈}        ¬ 𝜑

Theoremptcmplem5 21841* Lemma for ptcmp 21843. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))       (𝜑 → (∏t𝐹) ∈ Comp)

Theoremptcmpg 21842 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 21843). (Contributed by Mario Carneiro, 27-Aug-2015.)
𝐽 = (∏t𝐹)    &   𝑋 = 𝐽       ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)

Theoremptcmp 21843 Tychonoff's theorem: The product of compact spaces is compact. The proof uses the Axiom of Choice. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐴𝑉𝐹:𝐴⟶Comp) → (∏t𝐹) ∈ Comp)

12.2.5  Extension by continuity

Syntaxccnext 21844 Extend class notation with the continuous extension operation.
class CnExt

Definitiondf-cnext 21845* Define the continuous extension of a given function. (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ ( 𝑘pm 𝑗) ↦ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))

Theoremcnextval 21846* The function applying continuous extension to a given function 𝑓. (Contributed by Thierry Arnoux, 1-Dec-2017.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ ( 𝐾pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))

Theoremcnextfval 21847* The continuous extension of a given function 𝐹. (Contributed by Thierry Arnoux, 1-Dec-2017.)
𝑋 = 𝐽    &   𝐵 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))

Theoremcnextrel 21848 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:𝐴𝐵𝐴𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹))

Theoremcnextfun 21849 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:𝐴𝐵𝐴𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))

Theoremcnextfvval 21850* 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    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)       ((𝜑𝑋𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Theoremcnextf 21851* Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
𝐶 = 𝐽    &   𝐵 = 𝐾    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)       (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶𝐵)

Theoremcnextcn 21852* 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    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)    &   (𝜑𝐾 ∈ Reg)       (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))

Theoremcnextfres1 21853* 𝐹 and its extension by continuity agree on the domain of 𝐹. (Contributed by Thierry Arnoux, 17-Jan-2018.)
𝐶 = 𝐽    &   𝐵 = 𝐾    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)    &   (𝜑𝐾 ∈ Reg)    &   (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))       (𝜑 → (((𝐽CnExt𝐾)‘𝐹) ↾ 𝐴) = 𝐹)

Theoremcnextfres 21854 𝐹 and its extension by continuity agree on the domain of 𝐹. (Contributed by Thierry Arnoux, 29-Aug-2020.)
𝐶 = 𝐽    &   𝐵 = 𝐾    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐴𝐶)    &   (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))    &   (𝜑𝑋𝐴)       (𝜑 → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = (𝐹𝑋))

12.2.6  Topological groups

Syntaxctmd 21855 Extend class notation with the class of all topological monoids.
class TopMnd

Syntaxctgp 21856 Extend class notation with the class of all topological groups.
class TopGrp

Definitiondf-tmd 21857* 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](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}

Definitiondf-tgp 21858* 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](invg𝑓) ∈ (𝑗 Cn 𝑗)}

Theoremistmd 21859 The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐹 = (+𝑓𝐺)    &   𝐽 = (TopOpen‘𝐺)       (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Theoremtmdmnd 21860 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)

Theoremtmdtps 21861 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)

Theoremistgp 21862 The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))

Theoremtgpgrp 21863 A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Theoremtgptmd 21864 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

Theoremtgptps 21865 A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)

Theoremtmdtopon 21866 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))

Theoremtgptopon 21867 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))

Theoremtmdcn 21868 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 = (+𝑓𝐺)       (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Theoremtgpcn 21869 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 = (+𝑓𝐺)       (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Theoremtgpinv 21870 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 = (invg𝐺)       (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))

Theoremgrpinvhmeo 21871 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 = (invg𝐺)       (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽))

Theoremcnmpt1plusg 21872* Continuity of the group sum; analogue of cnmpt12f 21450 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ TopMnd)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))       (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))

Theoremcnmpt2plusg 21873* Continuity of the group sum; analogue of cnmpt22f 21459 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ TopMnd)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑𝐿 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))

Theoremtmdcn2 21874* Write out the definition of continuity of +g explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &    + = (+g𝐺)       (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))

Theoremtgpsubcn 21875 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)    &    = (-g𝐺)       (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Theoremistgp2 21876 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)    &    = (-g𝐺)       (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Theoremtmdmulg 21877* In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &    · = (.g𝐺)    &   𝐵 = (Base‘𝐺)       ((𝐺 ∈ TopMnd ∧ 𝑁 ∈ ℕ0) → (𝑥𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽))

Theoremtgpmulg 21878* In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &    · = (.g𝐺)    &   𝐵 = (Base‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑥𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽))

Theoremtgpmulg2 21879 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 22600 to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &    · = (.g𝐺)       (𝐺 ∈ TopGrp → · ∈ ((𝒫 ℤ ×t 𝐽) Cn 𝐽))

Theoremtmdgsum 21880* 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 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))

Theoremtmdgsum2 21881* 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)    &    · = (.g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋𝐵)    &   (𝜑 → ((#‘𝐴) · 𝑋) ∈ 𝑈)       (𝜑 → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))

Theoremoppgtmd 21882 The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd)

Theoremoppgtgp 21883 The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ TopGrp → 𝑂 ∈ TopGrp)

Theoremdistgp 21884 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)

Theoremindistgp 21885 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)

Theoremsymgtgp 21886 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉𝐺 ∈ TopGrp)

Theoremtmdlactcn 21887* 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 = (𝑥𝑋 ↦ (𝐴 + 𝑥))    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))

Theoremtgplacthmeo 21888* 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 = (𝑥𝑋 ↦ (𝐴 + 𝑥))    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))

Theoremsubmtmd 21889 A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)

Theoremsubgtgp 21890 A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)

Theoremsubgntr 21891 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21893, 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)) → 𝑆𝐽)

Theoremopnsubg 21892 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) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Theoremclssubg 21893 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))

Theoremclsnsg 21894 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))

Theoremcldsubg 21895 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) ↔ 𝑆𝐽))

Theoremtgpconncompeqg 21896* 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 = (0g𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}    &    = (𝐺 ~QG 𝑆)       ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})

Theoremtgpconncomp 21897* The identity component, the connected component containing the identity element, is a closed (conncompcld 21218) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))

Theoremtgpconncompss 21898* The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇𝐽) → 𝑆𝑇)

Theoremghmcnp 21899 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) ↔ (𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾))))

Theoremsnclseqg 21900 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 = (0g𝐺)    &    = (𝐺 ~QG 𝑆)    &   𝑆 = ((cls‘𝐽)‘{ 0 })       ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘{𝐴}))

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