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Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-perf 16701 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 16702* 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 16703* 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 16704 The predicate " is a limit point of ." (Contributed by NM, 10-Feb-2007.)

Theoremlpsscls 16705 The limits points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007.)

Theoremlpss 16706 The limits points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007.)

Theoremlpdifsn 16707 is a limit point of iff it is a limit point of . (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremlpss3 16708 Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremislp2 16709* 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 16710 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 16711 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 16712 A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)

Theoremcldlp 16713 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 16714 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremisperf2 16715 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremisperf3 16716* A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperflp 16717 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperfi 16718 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperftop 16719 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
Perf

11.1.7  Subspace topologies

Theoremrestrcl 16720 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremrestbas 16721 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 16722 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 16723 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 16724 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrestuni 16725 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
t

Theoremstoig 16726 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 16727 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
t t t

Theoremrestabs 16728 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 16729 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 16730 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theoremresttopon2 16731 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrest0 16732 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 16733 The only subspace topology induced by the topology . (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestsn2 16734 The subspace topology induced by a singleton. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn t

Theoremrestcld 16735* 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 16736 A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
t

Theoremrestcldr 16737 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 16738 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 16739 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 16740 The if is open, then is open in iff it is an open subset of . (Contributed by Mario Carneiro, 2-Mar-2015.)
t

Theoremrestdis 16741 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremrestfpw 16742 The restriction of the set of finite subsets of is the set of finite subsets of . (Contributed by Mario Carneiro, 18-Sep-2015.)
t

Theoremrestcls 16743 A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestntr 16744 An interior in a subspace topology. Willard in General Topology says that there is no analog of restcls 16743 for interiors. In some sense, that is true. (Contributed by Jeff Hankins, 23-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestlp 16745 The limit points of a subset restrict naturally in a subspace. (Contributed by Mario Carneiro, 25-Dec-2016.)
t

Theoremrestperf 16746 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 16747 An open subset of a perfect space is perfect. (Contributed by Mario Carneiro, 25-Dec-2016.)
t        Perf Perf

Theoremresstopn 16748 The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.)
s               t

Theoremresstps 16749 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 16750* Lemma for ordtbas 16754. In a total order, unbounded-above intervals are closed under intersection. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtval 16751* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtuni 16752* Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas2 16753* Lemma for ordtbas 16754. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremordtbas 16754* 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 16755 Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremordtopn1 16756* An upward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn2 16757* A downward ray is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtopn3 16758* An open interval is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld1 16759* A downward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld2 16760* An upward ray is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcld3 16761* An closed interval is closed. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordttop 16762 The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtcnv 16763 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop ordTop

Theoremordtrest 16764 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.)
ordTop ordTopt

Theoremordtrest2lem 16765* Lemma for ordtrest2 16766. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremordtrest2 16766* An interval-closed set 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 that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop ordTopt

Theoremletopon 16767 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop TopOn

Theoremletop 16768 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremletopuni 16769 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremxrstopn 16770 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
ordTop

Theoremxrstps 16771 The extended real number structure is a topological space. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremleordtvallem1 16772* Lemma for leordtval 16775. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtvallem2 16773* Lemma for leordtval 16775. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremleordtval2 16774 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremleordtval 16775 The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiccordt 16776 A closed interval is closed in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiocpnfordt 16777 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremicomnfordt 16778 An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremiooordt 16779 An open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremreordt 16780 The real numbers are an open set in the topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremlecldbas 16781 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.)
ordTop

Theorempnfnei 16782* A neighborhood of contains an unbounded interval based at a real number. Together with xrtgioo 18144 (which describes neighborhoods of ) and mnfnei 16783, this gives all "negative" topological information ensuring that it is not too fine (and of course iooordt 16779 and similar ensure that it has all the sets we want). (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremmnfnei 16783* A neighborhood of contains an unbounded interval based at a real number. (Contributed by Mario Carneiro, 3-Sep-2015.)
ordTop

Theoremordtrestixx 16784* 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.)
ordTop t ordTop

Theoremordtresticc 16785 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 ordTop

11.1.9  Limits and Continuity in topological spaces

Syntaxccn 16786 Extend class notation with the set of continuous functions between topologies.

Syntaxccnp 16787 Extend class notation with the set of functions between topologies continuous at a point.

Syntaxclm 16788 Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.

Definitiondf-cn 16789* 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 16797 for the predicate form. (Contributed by NM, 17-Oct-2006.)

Definitiondf-cnp 16790* 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.)

Definitiondf-lm 16791* Define a function on topologies whose value is the convergence relation for the space. Although 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 5454, and we use the otherwise unnecessary conjunct to ensure that. (Contributed by NM, 7-Sep-2006.)

Theoremlmrel 16792 The topological space convergence relation is a relation. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Theoremlmrcl 16793 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)

Theoremlmfval 16794* The relation "sequence converges to point " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnfval 16795* The set of all continuous functions from topology to topology . (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpfval 16796* The function mapping the points in a topology to the set of all functions from to topology continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremiscn 16797* The predicate " is a continuous function from topology to topology ." Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnpval 16798* The set of all functions from topology to topology that are continuous at a point . (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
TopOn TopOn

Theoremiscnp 16799* The predicate " is a continuous function from topology to topology at point ." Based on Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremiscn2 16800* The predicate " is a continuous function from topology to topology ." Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)

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