P. 241 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isflf 24001*	The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))
Theorem	flfelbas 24002	A limit point of a function is in the topological space. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) → 𝐴 ∈ 𝑋)
Theorem	flffbas 24003*	Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐿 = (𝑌filGen𝐵)    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))))
Theorem	flftg 24004*	Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (topGen‘𝐵)    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))
Theorem	hausflf 24005*	If a function has its values in a Hausdorff space, then it has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
Theorem	hausflf2 24006	If a convergent function has its values in a Hausdorff space, then it has a unique limit. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1o)
Theorem	cnpflfi 24007	Forward direction of cnpflf 24009. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
Theorem	cnpflf2 24008	𝐹 is continuous at point 𝐴 iff a limit of 𝐹 when 𝑥 tends to 𝐴 is (𝐹‘𝐴). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
⊢ 𝐿 = ((nei‘𝐽)‘{𝐴})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))))
Theorem	cnpflf 24009*	Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
Theorem	cnflf 24010*	A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
Theorem	cnflf2 24011*	A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐹 “ (𝐽 fLim 𝑓)) ⊆ ((𝐾 fLimf 𝑓)‘𝐹))))
Theorem	flfcnp 24012	A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)))
Theorem	lmflf 24013	The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍))    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹)))
Theorem	txflf 24014*	Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍))    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)    &   ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)    ⇒   ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺))))
Theorem	flfcnp2 24015*	The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐴)))    &   ⊢ (𝜑 → 𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐵)))    &   ⊢ (𝜑 → 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))    ⇒   ⊢ (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))
Theorem	fclsval 24016*	The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
Theorem	isfcls 24017*	A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
Theorem	fclsfil 24018	Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))
Theorem	fclstop 24019	Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
Theorem	fclstopon 24020	Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
Theorem	isfcls2 24021*	A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
Theorem	fclsopn 24022*	Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
Theorem	fclsopni 24023	An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅)
Theorem	fclselbas 24024	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) → 𝐴 ∈ 𝑋)
Theorem	fclsneii 24025	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}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∩ 𝑆) ≠ ∅)
Theorem	fclssscls 24026	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.)
⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Theorem	fclsnei 24027*	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) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)))
Theorem	supnfcls 24028*	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 24029*	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 24030	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 24031	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 24032	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 24033*	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 24034	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 24035*	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 24036*	A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 24035, 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 24037	Forward direction of fclscmp 24038. 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 24038*	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 24039	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 24040*	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 24041	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 24042*	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 24043*	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 24044	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 24045	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 24046	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 24047	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 24048	Lemma for cnpfcf 24049. 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 24049*	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 24050*	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 24051	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 24052*	Lemma for alexsub 24053. (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 24053*	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 24059 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 24054*	Biconditional form of the Alexander Subbase Theorem alexsub 24053. (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 24055*	Lemma for alexsubALT 24059. 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 24056*	Lemma for alexsubALT 24059. 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 24057*	Lemma for alexsubALT 24059. 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 24058*	Lemma for alexsubALT 24059. 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 24059*	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 24060*	Lemma for ptcmp 24066. (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 24061*	Lemma for ptcmp 24066. (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 24062*	Lemma for ptcmp 24066. (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 24063*	Lemma for ptcmp 24066. (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 24064*	Lemma for ptcmp 24066. (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 24065	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 24066). (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 24066	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 24067	Extend class notation with the continuous extension operation.
class CnExt
Definition	df-cnext 24068*	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 24069*	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 24070*	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 24071	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 24072	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 24073*	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 24074*	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 24075*	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 24076*	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 24077	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 24078	Extend class notation with the class of all topological monoids.
class TopMnd
Syntax	ctgp 24079	Extend class notation with the class of all topological groups.
class TopGrp
Definition	df-tmd 24080*	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 24081*	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 24082	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 24083	A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
Theorem	tmdtps 24084	A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
Theorem	istgp 24085	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 24086	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 24087	A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
Theorem	tgptps 24088	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 24089	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 24090	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 24091	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 24092	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 24093	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 24094	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 24095*	Continuity of the group sum; analogue of cnmpt12f 23674 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 24096*	Continuity of the group sum; analogue of cnmpt22f 23683 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 24097*	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 24098	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 24099	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 24100*	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))