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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremislpi 21001 A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ (𝑃 ∈ ((cls‘𝐽)‘𝑆) ∧ ¬ 𝑃𝑆)) → 𝑃 ∈ ((limPt‘𝐽)‘𝑆))

Theoremcldlp 21002 A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((limPt‘𝐽)‘𝑆) ⊆ 𝑆))

Theoremisperf 21003 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ((limPt‘𝐽)‘𝑋) = 𝑋))

Theoremisperf2 21004 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))

Theoremisperf3 21005* A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))

Theoremperflp 21006 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       (𝐽 ∈ Perf → ((limPt‘𝐽)‘𝑋) = 𝑋)

Theoremperfi 21007 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝑋 = 𝐽       ((𝐽 ∈ Perf ∧ 𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)

Theoremperftop 21008 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐽 ∈ Perf → 𝐽 ∈ Top)

12.1.7  Subspace topologies

Theoremrestrcl 21009 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))

Theoremrestbas 21010 A subspace topology basis is a basis. 𝑌 is normally a subset of the base set of 𝐽. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐵 ∈ TopBases → (𝐵t 𝐴) ∈ TopBases)

Theoremtgrest 21011 A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))

Theoremresttop 21012 A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. 𝐴 is normally a subset of the base set of 𝐽. (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)

Theoremresttopon 21013 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))

Theoremrestuni 21014 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))

Theoremstoig 21015 The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), (𝐽t 𝐴)⟩} ∈ TopSp)

Theoremrestco 21016 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (𝐽t (𝐴𝐵)))

Theoremrestabs 21017 Equivalence of being a subspace of a subspace and being a subspace of the original. (Contributed by Jeff Hankins, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
((𝐽𝑉𝑆𝑇𝑇𝑊) → ((𝐽t 𝑇) ↾t 𝑆) = (𝐽t 𝑆))

Theoremrestin 21018 When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽       ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = (𝐽t (𝐴𝑋)))

Theoremrestuni2 21019 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴𝑋) = (𝐽t 𝐴))

Theoremresttopon2 21020 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ (TopOn‘(𝐴𝑋)))

Theoremrest0 21021 The subspace topology induced by the topology 𝐽 on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
(𝐽 ∈ Top → (𝐽t ∅) = {∅})

Theoremrestsn 21022 The only subspace topology induced by the topology {∅}. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Theoremrestsn2 21023 The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) = 𝒫 {𝐴})

Theoremrestcld 21024* A closed set of a subspace topology is a closed set of the original topology intersected with the subset. (Contributed by FL, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐴 ∈ (Clsd‘(𝐽t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥𝑆)))

Theoremrestcldi 21025 A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑋 = 𝐽       ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))

Theoremrestcldr 21026 A set which is closed in the subspace topology induced by a closed set is closed in the original topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))

Theoremrestopnb 21027 If 𝐵 is an open subset of the subspace base set 𝐴, then any subset of 𝐵 is open iff it is open in 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
(((𝐽 ∈ Top ∧ 𝐴𝑉) ∧ (𝐵𝐽𝐵𝐴𝐶𝐵)) → (𝐶𝐽𝐶 ∈ (𝐽t 𝐴)))

Theoremssrest 21028 If 𝐾 is a finer topology than 𝐽, then the subspace topologies induced by 𝐴 maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐾𝑉𝐽𝐾) → (𝐽t 𝐴) ⊆ (𝐾t 𝐴))

Theoremrestopn2 21029 The if 𝐴 is open, then 𝐵 is open in 𝐴 iff it is an open subset of 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝐵 ∈ (𝐽t 𝐴) ↔ (𝐵𝐽𝐵𝐴)))

Theoremrestdis 21030 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝐴𝑉𝐵𝐴) → (𝒫 𝐴t 𝐵) = 𝒫 𝐵)

Theoremrestfpw 21031 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.)
((𝐴𝑉𝐵𝐴) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) = (𝒫 𝐵 ∩ Fin))

Theoremneitr 21032 The neighborhood of a trace is the trace of the neighborhood. (Contributed by Thierry Arnoux, 17-Jan-2018.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((nei‘(𝐽t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴))

Theoremrestcls 21033 A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽    &   𝐾 = (𝐽t 𝑌)       ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Theoremrestntr 21034 An interior in a subspace topology. Willard in General Topology says that there is no analogue of restcls 21033 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 = (𝐽t 𝑌)       ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋𝑌))) ∩ 𝑌))

Theoremrestlp 21035 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽    &   𝐾 = (𝐽t 𝑌)       ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌))

Theoremrestperf 21036 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 = (𝐽t 𝑌)       ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌)))

Theoremperfopn 21037 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑋 = 𝐽    &   𝐾 = (𝐽t 𝑌)       ((𝐽 ∈ Perf ∧ 𝑌𝐽) → 𝐾 ∈ Perf)

Theoremresstopn 21038 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐾s 𝐴)    &   𝐽 = (TopOpen‘𝐾)       (𝐽t 𝐴) = (TopOpen‘𝐻)

Theoremresstps 21039 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 ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ TopSp)

12.1.8  Order topology

Theoremordtbaslem 21040* Lemma for ordtbas 21044. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})       (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)

Theoremordtval 21041* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})       (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (𝐴𝐵)))))

Theoremordtuni 21042* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})       (𝑅𝑉𝑋 = ({𝑋} ∪ (𝐴𝐵)))

Theoremordtbas2 21043* Lemma for ordtbas 21044. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅    &   𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})    &   𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})       (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))

Theoremordtbas 21044* 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 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})    &   𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})    &   𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})       (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶))

Theoremordttopon 21045 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))

Theoremordtopn1 21046* An upward ray (𝑃, +∞) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))

Theoremordtopn2 21047* A downward ray (-∞, 𝑃) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))

Theoremordtopn3 21048* An open interval (𝐴, 𝐵) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝐴𝑋𝐵𝑋) → {𝑥𝑋 ∣ (¬ 𝑥𝑅𝐴 ∧ ¬ 𝐵𝑅𝑥)} ∈ (ordTop‘𝑅))

Theoremordtcld1 21049* A downward ray (-∞, 𝑃] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋𝑥𝑅𝑃} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordtcld2 21050* An upward ray [𝑃, +∞) is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋𝑃𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordtcld3 21051* A closed interval [𝐴, 𝐵] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝐴𝑋𝐵𝑋) → {𝑥𝑋 ∣ (𝐴𝑅𝑥𝑥𝑅𝐵)} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordttop 21052 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅𝑉 → (ordTop‘𝑅) ∈ Top)

Theoremordtcnv 21053 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅 ∈ PosetRel → (ordTop‘𝑅) = (ordTop‘𝑅))

Theoremordtrest 21054 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 ∧ 𝐴𝑉) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))

Theoremordtrest2lem 21055* Lemma for ordtrest2 21056. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅    &   (𝜑𝑅 ∈ TosetRel )    &   (𝜑𝐴𝑋)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)       (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))

Theoremordtrest2 21056* 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 , but in other sets like there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = dom 𝑅    &   (𝜑𝑅 ∈ TosetRel )    &   (𝜑𝐴𝑋)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)       (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))

Theoremletopon 21057 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(ordTop‘ ≤ ) ∈ (TopOn‘ℝ*)

Theoremletop 21058 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(ordTop‘ ≤ ) ∈ Top

Theoremletopuni 21059 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
* = (ordTop‘ ≤ )

Theoremxrstopn 21060 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
(ordTop‘ ≤ ) = (TopOpen‘ℝ*𝑠)

Theoremxrstps 21061 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 ∈ TopSp

Theoremleordtvallem1 21062* Lemma for leordtval 21065. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))       𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦𝑥})

Theoremleordtvallem2 21063* Lemma for leordtval 21065. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))    &   𝐵 = ran (𝑥 ∈ ℝ* ↦ (-∞[,)𝑥))       𝐵 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑥𝑦})

Theoremleordtval2 21064 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‘(𝐴𝐵)))

Theoremleordtval 21065 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‘((𝐴𝐵) ∪ 𝐶))

Theoremiccordt 21066 A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴[,]𝐵) ∈ (Clsd‘(ordTop‘ ≤ ))

Theoremiocpnfordt 21067 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴(,]+∞) ∈ (ordTop‘ ≤ )

Theoremicomnfordt 21068 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(-∞[,)𝐴) ∈ (ordTop‘ ≤ )

Theoremiooordt 21069 An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴(,)𝐵) ∈ (ordTop‘ ≤ )

Theoremreordt 21070 The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ℝ ∈ (ordTop‘ ≤ )

Theoremlecldbas 21071 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))

Theorempnfnei 21072* A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 22656 (which describes neighborhoods of ) and mnfnei 21073, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 21069 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)

Theoremmnfnei 21073* 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)

Theoremordtrestixx 21074* 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*    &   ((𝑥𝐴𝑦𝐴) → (𝑥[,]𝑦) ⊆ 𝐴)       ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))

Theoremordtresticc 21075 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))))

12.1.9  Limits and continuity in topological spaces

Syntaxccn 21076 Extend class notation with the class of continuous functions between topologies.
class Cn

Syntaxccnp 21077 Extend class notation with the class of functions between topologies continuous at a given point.
class CnP

Syntaxclm 21078 Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.
class 𝑡

Definitiondf-cn 21079* 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 21087 for the predicate form. (Contributed by NM, 17-Oct-2006.)
Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})

Definitiondf-cnp 21080* 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 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 ((𝑓𝑥) ∈ 𝑦 → ∃𝑔𝑗 (𝑥𝑔 ∧ (𝑓𝑔) ⊆ 𝑦))}))

Definitiondf-lm 21081* 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 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})

Theoremlmrel 21082 The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
Rel (⇝𝑡𝐽)

Theoremlmrcl 21083 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
(𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ Top)

Theoremlmfval 21084* 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) → (⇝𝑡𝐽) = {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (𝑋pm ℂ) ∧ 𝑥𝑋 ∧ ∀𝑢𝐽 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})

Theoremcnfval 21085* 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 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})

Theoremcnpfval 21086* 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) = (𝑥𝑋 ↦ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑤𝐾 ((𝑓𝑥) ∈ 𝑤 → ∃𝑣𝐽 (𝑥𝑣 ∧ (𝑓𝑣) ⊆ 𝑤))}))

Theoremiscn 21087* The predicate "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:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))

Theoremcnpval 21088* 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) ∧ 𝑃𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 ((𝑓𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑓𝑥) ⊆ 𝑦))})

Theoremiscnp 21089* The predicate "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) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))

Theoremiscn2 21090* The predicate "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:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))

Theoremiscnp2 21091* The predicate "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 ∧ 𝑃𝑋) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))

Theoremcntop1 21092 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)

Theoremcntop2 21093 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)

Theoremcnptop1 21094 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)

Theoremcnptop2 21095 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)

Theoremiscnp3 21096* The predicate "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) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥𝑥 ⊆ (𝐹𝑦))))))

Theoremcnprcl 21097 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽       (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃𝑋)

Theoremcnf 21098 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:𝑋𝑌)

Theoremcnpf 21099 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:𝑋𝑌)

Theoremcnpcl 21100 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) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)

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