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Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmreclatdemo 17101 The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclat 14554. (Contributed by Stefan O'Rear, 31-Jan-2015.)
toInc

11.1.5  Neighborhoods

Syntaxcnei 17102 Extend class notation with neighborhood relation for topologies.

Definitiondf-nei 17103* Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)

Theoremneifval 17104* The neighborhood function on the subsets of a topology's base set. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremneif 17105 The neighborhood function is a function of the subsets of a topology's base set. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremneiss2 17106 A set with a neighborhood is a subset of the topology's base set. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)

Theoremneival 17107* The set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremisnei 17108* The predicate " is a neighborhood of ." (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremneiint 17109 An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremisneip 17110* The predicate " is a neighborhood of point ." (Contributed by NM, 26-Feb-2007.)

Theoremneii1 17111 A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)

Theoremneisspw 17112 The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)

Theoremneii2 17113* Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)

Theoremneiss 17114 Any neighborhood of a set is also a neighborhood of any subset . Theorem of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)

Theoremssnei 17115 A set is included in its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3 . (Contributed by FL, 16-Nov-2006.)

Theoremelnei 17116 A point belongs to any of its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)

Theorem0nnei 17117 The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)

Theoremneips 17118* A neighborhood of a set is a neighborhood of every point in the set. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.)

Theoremopnneissb 17119 An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)

Theoremopnssneib 17120 Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)

Theoremssnei2 17121 Any subset of containing a neigborhood of a set is a neighborhood of this set. Proposition Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)

Theoremneindisj 17122 Any neighborhood of an element in the closure of a subset intersects the subset. Part of proof of Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)

Theoremopnneiss 17123 An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)

Theoremopnneip 17124 An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)

Theoremopnnei 17125* A set is open iff it is a neighborhood of all its points. ( Contributed by Jeff Hankins, 15-Sep-2009.) (Contributed by NM, 16-Sep-2009.)

Theoremtpnei 17126 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 17123. (Contributed by FL, 2-Oct-2006.)

Theoremneiuni 17127 The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)

Theoremneindisj2 17128* A point belongs to the closure of a set iff every neighborhood of meets . (Contributed by FL, 15-Sep-2013.)

Theoremtopssnei 17129 A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)

Theoreminnei 17130 The intersection of two neighborhoods of a set is also a neighborhood of the set. Proposition Vii of [BourbakiTop1] p. I.3 . (Contributed by FL, 28-Sep-2006.)

Theoremopnneiid 17131 Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)

Theoremneissex 17132* For any neighborhood of , there is a neighborhood of such that is a neighborhood of all subsets of . Proposition Viv of [BourbakiTop1] p. I.3 . (Contributed by FL, 2-Oct-2006.)

Theorem0nei 17133 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)

Theoremneipeltop 17134* Lemma for neiptopreu 17138 (Contributed by Thierry Arnoux, 6-Jan-2018.)

Theoremneiptopuni 17135* Lemma for neiptopreu 17138 (Contributed by Thierry Arnoux, 6-Jan-2018.)

Theoremneiptoptop 17136* Lemma for neiptopreu 17138 (Contributed by Thierry Arnoux, 7-Jan-2018.)

Theoremneiptopnei 17137* Lemma for neiptopreu 17138 (Contributed by Thierry Arnoux, 7-Jan-2018.)

Theoremneiptopreu 17138* If, to each element of a set , we associate a set fulfilling the properties Vi, Vii, Viii and property Viv of [BourbakiTop1] p. I.2. , corresponding to ssnei 17115, innei 17130, elnei 17116 and neissex 17132, then there is a unique topology such that for any point , is the set of neighborhoods of . Proposition 2 of [BourbakiTop1] p. I.3. This can be used to build a topology from a set of neighborhoods. Note that the additional condition that is a neighborhood of all points was added. (Contributed by Thierry Arnoux, 6-Jan-2018.)
TopOn

11.1.6  Limit points and perfect sets

Syntaxclp 17139 Extend class notation with the limit point function for topologies.

Syntaxcperf 17140 Extend class notation with the class of all perfect spaces.
Perf

Definitiondf-lp 17141* Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 17144. (Contributed by NM, 10-Feb-2007.)

Definitiondf-perf 17142 Define the class of all perfect spaces. A perfect space is one for which every point in the set is a limit point of the whole space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremlpfval 17143* The limit point function on the subsets of a topology's base set. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremlpval 17144* The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremislp 17145 The predicate " is a limit point of ." (Contributed by NM, 10-Feb-2007.)

Theoremlpsscls 17146 The limit points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007.)

Theoremlpss 17147 The limit points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007.)

Theoremlpdifsn 17148 is a limit point of iff it is a limit point of . (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremlpss3 17149 Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremislp2 17150* The predicate " is a limit point of ," in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.)

Theoremmaxlp 17151 A point is a limit point of the whole space iff the singleton of the point is not open. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremclslp 17152 The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)

Theoremislpi 17153 A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)

Theoremcldlp 17154 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.)

Theoremisperf 17155 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremisperf2 17156 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremisperf3 17157* A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperflp 17158 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperfi 17159 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperftop 17160 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
Perf

11.1.7  Subspace topologies

Theoremrestrcl 17161 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremrestbas 17162 A subspace topology basis is a basis. is normally a subset of the base set of . (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremtgrest 17163 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.)
t t

Theoremresttop 17164 A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. is normally a subset of the base set of . (Contributed by FL, 15-Apr-2007.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremresttopon 17165 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrestuni 17166 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
t

Theoremstoig 17167 The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.)
TopSet t

Theoremrestco 17168 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
t t t

Theoremrestabs 17169 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.)
t t t

Theoremrestin 17170 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.)
t t

Theoremrestuni2 17171 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theoremresttopon2 17172 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrest0 17173 The subspace topology induced by the topology on the empty set. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremrestsn 17174 The only subspace topology induced by the topology . (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestsn2 17175 The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn t

Theoremrestcld 17176* 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.)
t

Theoremrestcldi 17177 A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
t

Theoremrestcldr 17178 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.)
t

Theoremrestopnb 17179 If is an open subset of the subspace base set , then any subset of is open iff it is open in . (Contributed by Mario Carneiro, 2-Mar-2015.)
t

Theoremssrest 17180 If is a finer topology than , then the subspace topologies induced by maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
t t

Theoremrestopn2 17181 The if is open, then is open in iff it is an open subset of . (Contributed by Mario Carneiro, 2-Mar-2015.)
t

Theoremrestdis 17182 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremrestfpw 17183 The restriction of the set of finite subsets of is the set of finite subsets of . (Contributed by Mario Carneiro, 18-Sep-2015.)
t

Theoremneitr 17184 The neighborhood of a trace is the trace of the neighborhood. (Contributed by Thierry Arnoux, 17-Jan-2018.)
t t

Theoremrestcls 17185 A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestntr 17186 An interior in a subspace topology. Willard in General Topology says that there is no analog of restcls 17185 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestlp 17187 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
t

Theoremrestperf 17188 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.)
t        Perf

Theoremperfopn 17189 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        Perf Perf

Theoremresstopn 17190 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
s               t

Theoremresstps 17191 A restricted topological space is a topological space. Note that this theorem would not be true if was defined directly in terms of the TopSet slot instead of the derived function. (Contributed by Mario Carneiro, 13-Aug-2015.)
s

11.1.8  Order topology

Theoremordtbaslem 17192* Lemma for ordtbas 17196. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtval 17193* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtuni 17194* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas2 17195* Lemma for ordtbas 17196. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas 17196* 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.)

Theoremordttopon 17197 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremordtopn1 17198* An upward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn2 17199* A downward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn3 17200* An open interval is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

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