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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremresstps 20801 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 20802* Lemma for ordtbas 20806. 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 20803* 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 20804* 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 20805* Lemma for ordtbas 20806. (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 20806* 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 20807 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))

Theoremordtopn1 20808* An upward ray (𝑃, +∞) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))

Theoremordtopn2 20809* A downward ray (-∞, 𝑃) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))

Theoremordtopn3 20810* An open interval (𝐴, 𝐵) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝐴𝑋𝐵𝑋) → {𝑥𝑋 ∣ (¬ 𝑥𝑅𝐴 ∧ ¬ 𝐵𝑅𝑥)} ∈ (ordTop‘𝑅))

Theoremordtcld1 20811* A downward ray (-∞, 𝑃] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋𝑥𝑅𝑃} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordtcld2 20812* An upward ray [𝑃, +∞) is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝑃𝑋) → {𝑥𝑋𝑃𝑅𝑥} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordtcld3 20813* A closed interval [𝐴, 𝐵] is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝑋 = dom 𝑅       ((𝑅𝑉𝐴𝑋𝐵𝑋) → {𝑥𝑋 ∣ (𝐴𝑅𝑥𝑥𝑅𝐵)} ∈ (Clsd‘(ordTop‘𝑅)))

Theoremordttop 20814 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝑅𝑉 → (ordTop‘𝑅) ∈ Top)

Theoremordtcnv 20815 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 20816 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 20817* Lemma for ordtrest2 20818. (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 20818* 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 20819 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(ordTop‘ ≤ ) ∈ (TopOn‘ℝ*)

Theoremletop 20820 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(ordTop‘ ≤ ) ∈ Top

Theoremletopuni 20821 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
* = (ordTop‘ ≤ )

Theoremxrstopn 20822 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
(ordTop‘ ≤ ) = (TopOpen‘ℝ*𝑠)

Theoremxrstps 20823 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 ∈ TopSp

Theoremleordtvallem1 20824* Lemma for leordtval 20827. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐴 = ran (𝑥 ∈ ℝ* ↦ (𝑥(,]+∞))       𝐴 = ran (𝑥 ∈ ℝ* ↦ {𝑦 ∈ ℝ* ∣ ¬ 𝑦𝑥})

Theoremleordtvallem2 20825* Lemma for leordtval 20827. (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 20826 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 20827 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 20828 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 20829 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 20830 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 20831 An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴(,)𝐵) ∈ (ordTop‘ ≤ )

Theoremreordt 20832 The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ℝ ∈ (ordTop‘ ≤ )

Theoremlecldbas 20833 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 20834* A neighborhood of +∞ contains an unbounded interval based at a real number. Together with xrtgioo 22417 (which describes neighborhoods of ) and mnfnei 20835, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 20831 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 20835* 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 20836* 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 20837 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 20838 Extend class notation with the set of continuous functions between topologies.
class Cn

Syntaxccnp 20839 Extend class notation with the set of functions between topologies continuous at a point.
class CnP

Syntaxclm 20840 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 20841* 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 20849 for the predicate form. (Contributed by NM, 17-Oct-2006.)
Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})

Definitiondf-cnp 20842* 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 20843* Define a function on topologies whose value is the convergence relation for the space. Although 𝑓 is typically a function from upper integers to the topological space, it doesn't have to be. Unfortunately, the value of the function must exist to use fvmpt 6191, and we use the otherwise unnecessary conjunct dom 𝑓 ⊆ ℂ to ensure that. (Contributed by NM, 7-Sep-2006.)
𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})

Theoremlmrel 20844 The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
Rel (⇝𝑡𝐽)

Theoremlmrcl 20845 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)
(𝐹(⇝𝑡𝐽)𝑃𝐽 ∈ Top)

Theoremlmfval 20846* 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 20847* 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 20848* 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 20849* 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 20850* 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 20851* 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 20852* 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 20853* 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 20854 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)

Theoremcntop2 20855 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)

Theoremcnptop1 20856 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)

Theoremcnptop2 20857 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)

Theoremiscnp3 20858* 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 20859 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 20860 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 20861 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 20862 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)

Theoremcnf2 20863 A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)

Theoremcnpf2 20864 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:𝑋𝑌)

Theoremcnprcl2 20865 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃𝑋)

Theoremtgcn 20866* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 = (topGen‘𝐵))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐵 (𝐹𝑦) ∈ 𝐽)))

Theoremtgcnp 20867* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 = (topGen‘𝐵))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝑃𝑋)       (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐵 ((𝐹𝑃) ∈ 𝑦 → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦)))))

Theoremsubbascn 20868* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐵𝑉)    &   (𝜑𝐾 = (topGen‘(fi‘𝐵)))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐵 (𝐹𝑦) ∈ 𝐽)))

Theoremssidcn 20869 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) ↔ 𝐾𝐽))

Theoremcnpimaex 20870* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.)
((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴𝐾 ∧ (𝐹𝑃) ∈ 𝐴) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝐴))

Theoremidcn 20871 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))

Theoremlmbr 20872* 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 us to use objects more general than sequences when convenient; see the comment in df-lm 20843. (Contributed by Mario Carneiro, 14-Nov-2013.)
(𝜑𝐽 ∈ (TopOn‘𝑋))       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑦 ∈ ran ℤ(𝐹𝑦):𝑦𝑢))))

Theoremlmbr2 20873* 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.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))

Theoremlmbrf 20874* 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 20873 presupposes that 𝐹 is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍𝑋)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝐴𝑢))))

Theoremlmconst 20875 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
𝑍 = (ℤ𝑀)       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑃𝑋𝑀 ∈ ℤ) → (𝑍 × {𝑃})(⇝𝑡𝐽)𝑃)

Theoremlmcvg 20876* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑃𝑈)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑈𝐽)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑈)

Theoremiscnp4 20877* The predicate "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) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝑃)})∃𝑥 ∈ ((nei‘𝐽)‘{𝑃})(𝐹𝑥) ⊆ 𝑦)))

Theoremcnpnei 20878* 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:𝑋𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹𝐴)})(𝐹𝑦) ∈ ((nei‘𝐽)‘{𝐴})))

Theoremcnima 20879 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)
((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝐾) → (𝐹𝐴) ∈ 𝐽)

Theoremcnco 20880 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)) → (𝐺𝐹) ∈ (𝐽 Cn 𝐿))

Theoremcnpco 20881 The composition of two continuous functions at point 𝑃 is a continuous function at point 𝑃. Proposition 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)‘(𝐹𝑃))) → (𝐺𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))

Theoremcnclima 20882 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)) → (𝐹𝐴) ∈ (Clsd‘𝐽))

Theoremiscncl 20883* A definition of a continuous 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:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))

Theoremcncls2i 20884 Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑌 = 𝐾       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → ((cls‘𝐽)‘(𝐹𝑆)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑆)))

Theoremcnntri 20885 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑌 = 𝐾       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑌) → (𝐹 “ ((int‘𝐾)‘𝑆)) ⊆ ((int‘𝐽)‘(𝐹𝑆)))

Theoremcnclsi 20886 Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑆𝑋) → (𝐹 “ ((cls‘𝐽)‘𝑆)) ⊆ ((cls‘𝐾)‘(𝐹𝑆)))

Theoremcncls2 20887* Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌((cls‘𝐽)‘(𝐹𝑥)) ⊆ (𝐹 “ ((cls‘𝐾)‘𝑥)))))

Theoremcncls 20888* 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:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐹 “ ((cls‘𝐽)‘𝑥)) ⊆ ((cls‘𝐾)‘(𝐹𝑥)))))

Theoremcnntr 20889* 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:𝑋𝑌 ∧ ∀𝑥 ∈ 𝒫 𝑌(𝐹 “ ((int‘𝐾)‘𝑥)) ⊆ ((int‘𝐽)‘(𝐹𝑥)))))

Theoremcnss1 20890 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) ∧ 𝐽𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))

Theoremcnss2 20891 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) ∧ 𝐿𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))

Theoremcncnpi 20892 A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))

Theoremcnsscnp 20893 The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽       (𝑃𝑋 → (𝐽 Cn 𝐾) ⊆ ((𝐽 CnP 𝐾)‘𝑃))

Theoremcncnp 20894* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))

Theoremcncnp2 20895* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (𝑋 ≠ ∅ → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))

Theoremcnnei 20896* Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)
𝑋 = 𝐽    &   𝑌 = 𝐾       ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑝𝑋𝑤 ∈ ((nei‘𝐾)‘{(𝐹𝑝)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑝})(𝐹𝑣) ⊆ 𝑤))

Theoremcnconst2 20897 A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))

Theoremcnconst 20898 A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐵𝑌𝐹:𝑋⟶{𝐵})) → 𝐹 ∈ (𝐽 Cn 𝐾))

Theoremcnrest 20899 Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)
𝑋 = 𝐽       ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))

Theoremcnrest2 20900 Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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