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Type | Label | Description |
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Statement | ||
Theorem | restdis 23201 | A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → (𝒫 𝐴 ↾t 𝐵) = 𝒫 𝐵) | ||
Theorem | restfpw 23202 | The restriction of the set of finite subsets of 𝐴 is the set of finite subsets of 𝐵. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) = (𝒫 𝐵 ∩ Fin)) | ||
Theorem | neitr 23203 | The neighborhood of a trace is the trace of the neighborhood. (Contributed by Thierry Arnoux, 17-Jan-2018.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴)) | ||
Theorem | restcls 23204 | A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (𝐽 ↾t 𝑌) ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) | ||
Theorem | restntr 23205 | An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 23204 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (𝐽 ↾t 𝑌) ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)) | ||
Theorem | restlp 23206 | The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (𝐽 ↾t 𝑌) ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌)) | ||
Theorem | restperf 23207 | Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (𝐽 ↾t 𝑌) ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌))) | ||
Theorem | perfopn 23208 | An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (𝐽 ↾t 𝑌) ⇒ ⊢ ((𝐽 ∈ Perf ∧ 𝑌 ∈ 𝐽) → 𝐾 ∈ Perf) | ||
Theorem | resstopn 23209 | The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ 𝐻 = (𝐾 ↾s 𝐴) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐽 ↾t 𝐴) = (TopOpen‘𝐻) | ||
Theorem | resstps 23210 | A restricted topological space is a topological space. Note that this theorem would not be true if TopSp was defined directly in terms of the TopSet slot instead of the TopOpen derived function. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ ((𝐾 ∈ TopSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ TopSp) | ||
Theorem | ordtbaslem 23211* | Lemma for ordtbas 23215. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 & ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⇒ ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴) | ||
Theorem | ordtval 23212* | Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 & ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) & ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⇒ ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))))) | ||
Theorem | ordtuni 23213* | Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 & ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) & ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵))) | ||
Theorem | ordtbas2 23214* | Lemma for ordtbas 23215. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 & ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) & ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) & ⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⇒ ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)) | ||
Theorem | ordtbas 23215* | In a total order, the finite intersections of the open rays generates the set of open intervals, but no more - these four collections form a subbasis for the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 & ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) & ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) & ⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⇒ ⊢ (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴 ∪ 𝐵))) = (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∪ 𝐶)) | ||
Theorem | ordttopon 23216 | Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋)) | ||
Theorem | ordtopn1 23217* | An upward ray (𝑃, +∞) is open. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅)) | ||
Theorem | ordtopn2 23218* | A downward ray (-∞, 𝑃) is open. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅)) | ||
Theorem | ordtopn3 23219* | An open interval (𝐴, 𝐵) is open. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ (¬ 𝑥𝑅𝐴 ∧ ¬ 𝐵𝑅𝑥)} ∈ (ordTop‘𝑅)) | ||
Theorem | ordtcld1 23220* | A downward ray (-∞, 𝑃] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝑥𝑅𝑃} ∈ (Clsd‘(ordTop‘𝑅))) | ||
Theorem | ordtcld2 23221* | An upward ray [𝑃, +∞) is closed. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ 𝑃𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅))) | ||
Theorem | ordtcld3 23222* | A closed interval [𝐴, 𝐵] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ (𝐴𝑅𝑥 ∧ 𝑥𝑅𝐵)} ∈ (Clsd‘(ordTop‘𝑅))) | ||
Theorem | ordttop 23223 | The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ Top) | ||
Theorem | ordtcnv 23224 | The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑅 ∈ PosetRel → (ordTop‘◡𝑅) = (ordTop‘𝑅)) | ||
Theorem | ordtrest 23225 | 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.) |
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴)) | ||
Theorem | ordtrest2lem 23226* | Lemma for ordtrest2 23227. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 & ⊢ (𝜑 → 𝑅 ∈ TosetRel ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴) ⇒ ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) | ||
Theorem | ordtrest2 23227* | 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.) |
⊢ 𝑋 = dom 𝑅 & ⊢ (𝜑 → 𝑅 ∈ TosetRel ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴) ⇒ ⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴)) | ||
Theorem | letopon 23228 | The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | ||
Theorem | letop 23229 | The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (ordTop‘ ≤ ) ∈ Top | ||
Theorem | letopuni 23230 | The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ ℝ* = ∪ (ordTop‘ ≤ ) | ||
Theorem | xrstopn 23231 | The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (ordTop‘ ≤ ) = (TopOpen‘ℝ*𝑠) | ||
Theorem | xrstps 23232 | The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ*𝑠 ∈ TopSp | ||
Theorem | leordtvallem1 23233* | Lemma for leordtval 23236. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) ⇒ ⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥}) | ||
Theorem | leordtvallem2 23234* | Lemma for leordtval 23236. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) & ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) ⇒ ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥 ≤ 𝑦}) | ||
Theorem | leordtval2 23235 | The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) & ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) ⇒ ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(𝐴 ∪ 𝐵))) | ||
Theorem | leordtval 23236 | The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞)) & ⊢ 𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥)) & ⊢ 𝐶 = ran (,) ⇒ ⊢ (ordTop‘ ≤ ) = (topGen‘((𝐴 ∪ 𝐵) ∪ 𝐶)) | ||
Theorem | iccordt 23237 | A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ )) | ||
Theorem | iocpnfordt 23238 | An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝐴(,]+∞) ∈ (ordTop‘ ≤ ) | ||
Theorem | icomnfordt 23239 | An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (-∞[,)𝐴) ∈ (ordTop‘ ≤ ) | ||
Theorem | iooordt 23240 | An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝐴(,)𝐵) ∈ (ordTop‘ ≤ ) | ||
Theorem | reordt 23241 | The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ ℝ ∈ (ordTop‘ ≤ ) | ||
Theorem | lecldbas 23242 | The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ* ∖ 𝑥)) ⇒ ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹)) | ||
Theorem | pnfnei 23243* | A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 24841 (which describes neighborhoods of ℝ) and mnfnei 23244, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 23240 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴) | ||
Theorem | mnfnei 23244* | A neighborhood of -∞ contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ ((𝐴 ∈ (ordTop‘ ≤ ) ∧ -∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (-∞[,)𝑥) ⊆ 𝐴) | ||
Theorem | ordtrestixx 23245* | The restriction of the less than order to an interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ 𝐴 ⊆ ℝ* & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) ⇒ ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) | ||
Theorem | ordtresticc 23246 | The restriction of the less than order to a closed interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ ((ordTop‘ ≤ ) ↾t (𝐴[,]𝐵)) = (ordTop‘( ≤ ∩ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) | ||
Syntax | ccn 23247 | Extend class notation with the class of continuous functions between topologies. |
class Cn | ||
Syntax | ccnp 23248 | Extend class notation with the class of functions between topologies continuous at a given point. |
class CnP | ||
Syntax | clm 23249 | Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space. |
class ⇝𝑡 | ||
Definition | df-cn 23250* | Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 23258 for the predicate form. (Contributed by NM, 17-Oct-2006.) |
⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗}) | ||
Definition | df-cnp 23251* | Define a function on two topologies whose value is the set of continuous mappings at a specified point in the first topology. Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) |
⊢ CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})) | ||
Definition | df-lm 23252* | Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although 𝑓 is typically a sequence (a function from an upperset of integers) with values in the topological space, it need not be. Note, however, that the limit property concerns only values at integers, so that the real-valued function (𝑥 ∈ ℝ ↦ (sin‘(π · 𝑥))) converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.) |
⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}) | ||
Theorem | lmrel 23253 | The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.) |
⊢ Rel (⇝𝑡‘𝐽) | ||
Theorem | lmrcl 23254 | Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.) |
⊢ (𝐹(⇝𝑡‘𝐽)𝑃 → 𝐽 ∈ Top) | ||
Theorem | lmfval 23255* | The relation "sequence 𝑓 converges to point 𝑦 " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → (⇝𝑡‘𝐽) = {〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (𝑋 ↑pm ℂ) ∧ 𝑥 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}) | ||
Theorem | cnfval 23256* | The set of all continuous functions from topology 𝐽 to topology 𝐾. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽}) | ||
Theorem | cnpfval 23257* | The function mapping the points in a topology 𝐽 to the set of all functions from 𝐽 to topology 𝐾 continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑥 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑤 ∈ 𝐾 ((𝑓‘𝑥) ∈ 𝑤 → ∃𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 ∧ (𝑓 “ 𝑣) ⊆ 𝑤))})) | ||
Theorem | iscn 23258* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾". Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | ||
Theorem | cnpval 23259* | The set of all functions from topology 𝐽 to topology 𝐾 that are continuous at a point 𝑃. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))}) | ||
Theorem | iscnp 23260* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))) | ||
Theorem | iscn2 23261* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾". Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | ||
Theorem | iscnp2 23262* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))) | ||
Theorem | cntop1 23263 | Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | ||
Theorem | cntop2 23264 | Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | ||
Theorem | cnptop1 23265 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top) | ||
Theorem | cnptop2 23266 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top) | ||
Theorem | iscnp3 23267* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃". (Contributed by NM, 15-May-2007.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))))) | ||
Theorem | cnprcl 23268 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋) | ||
Theorem | cnf 23269 | A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌) | ||
Theorem | cnpf 23270 | A continuous function at point 𝑃 is a mapping. (Contributed by FL, 17-Nov-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶𝑌) | ||
Theorem | cnpcl 23271 | The value of a continuous function from 𝐽 to 𝐾 at point 𝑃 belongs to the underlying set of topology 𝐾. (Contributed by FL, 27-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌) | ||
Theorem | cnf2 23272 | A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌) | ||
Theorem | cnpf2 23273 | A continuous function at point 𝑃 is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌) | ||
Theorem | cnprcl2 23274 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋) | ||
Theorem | tgcn 23275* | The continuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 = (topGen‘𝐵)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))) | ||
Theorem | tgcnp 23276* | The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 = (topGen‘𝐵)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))) | ||
Theorem | subbascn 23277* | The continuity predicate when the range is given by a subbasis for a topology. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐾 = (topGen‘(fi‘𝐵))) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))) | ||
Theorem | ssidcn 23278 | The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 ⊆ 𝐽)) | ||
Theorem | cnpimaex 23279* | Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) |
⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝐴) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝐴)) | ||
Theorem | idcn 23280 | A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽)) | ||
Theorem | lmbr 23281* | Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows to use objects more general than sequences when convenient; see the comment in df-lm 23252. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝐹 ↾ 𝑦):𝑦⟶𝑢)))) | ||
Theorem | lmbr2 23282* | Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) | ||
Theorem | lmbrf 23283* | Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. This version of lmbr2 23282 presupposes that 𝐹 is a function. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐴 ∈ 𝑢)))) | ||
Theorem | lmconst 23284 | A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡‘𝐽)𝑃) | ||
Theorem | lmcvg 23285* | Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑃 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑈) | ||
Theorem | iscnp4 23286* | The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾 at point 𝑃 " in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹 “ 𝑥) ⊆ 𝑦))) | ||
Theorem | cnpnei 23287* | A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}))) | ||
Theorem | cnima 23288 | An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.) |
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝐾) → (◡𝐹 “ 𝐴) ∈ 𝐽) | ||
Theorem | cnco 23289 | The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿)) | ||
Theorem | cnpco 23290 | The composition of a function 𝐹 continuous at 𝑃 with a function continuous at (𝐹‘𝑃) is continuous at 𝑃. Proposition 2 of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃)) | ||
Theorem | cnclima 23291 | A closed subset of the codomain of a continuous function has a closed preimage. (Contributed by NM, 15-Mar-2007.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝐴) ∈ (Clsd‘𝐽)) | ||
Theorem | iscncl 23292* | A characterization of a continuity function using closed sets. Theorem 1(d) of [BourbakiTop1] p. I.9. (Contributed by FL, 19-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽)))) | ||
Theorem | cncls2i 23293 | Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘(◡𝐹 “ 𝑆)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑆))) | ||
Theorem | cnntri 23294 | Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑌) → (◡𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑆))) | ||
Theorem | cnclsi 23295 | Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆 ⊆ 𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑆))) | ||
Theorem | cncls2 23296* | Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(◡𝐹 “ 𝑥)) ⊆ (◡𝐹 “ ((cls‘𝐾)‘𝑥))))) | ||
Theorem | cncls 23297* | Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹 “ 𝑥))))) | ||
Theorem | cnntr 23298* | Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(◡𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑥))))) | ||
Theorem | cnss1 23299 | If the topology 𝐾 is finer than 𝐽, then there are more continuous functions from 𝐾 than from 𝐽. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿)) | ||
Theorem | cnss2 23300 | If the topology 𝐾 is finer than 𝐽, then there are fewer continuous functions into 𝐾 than into 𝐽 from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿)) |
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