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Theorem List for Metamath Proof Explorer - 16801-16900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiscld3 16801 A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006.)

Theoremiscld4 16802 A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008.)

Theoremisopn3 16803 A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremclsidm 16804 The closure operation is idempotent. (Contributed by NM, 2-Oct-2007.)

Theoremntridm 16805 The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.)

Theoremclstop 16806 The closure of a topology's underlying set is entire set. (Contributed by NM, 5-Oct-2007.)

Theoremntrtop 16807 The interior of a topology's underlying set is entire set. (Contributed by NM, 12-Sep-2006.)

Theorem0ntr 16808 A subset with an empty interior cannot cover a whole (nonempty) topology. (Contributed by NM, 12-Sep-2006.)

Theoremclsss2 16809 If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.)

Theoremelcls 16810* Membership in a closure. Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 22-Feb-2007.)

Theoremelcls2 16811* Membership in a closure. (Contributed by NM, 5-Mar-2007.)

Theoremclsndisj 16812 Any open set containing a point that belongs to the closure of a subset intersects the subset. One direction of Theorem 6.5(a) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.)

Theoremntrcls0 16813 A subset whose closure has an empty interior also has an empty interior. (Contributed by NM, 4-Oct-2007.)

Theoremntreq0 16814* Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremcldmre 16815 The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Moore

Theoremmrccls 16816 Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls

Theoremcls0 16817 The closure of the empty set. (Contributed by NM, 2-Oct-2007.)

Theoremntr0 16818 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)

Theoremisopn3i 16819 An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremelcls3 16820* Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95. (Contributed by NM, 26-Feb-2007.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremopncldf1 16821* A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremopncldf2 16822* The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremopncldf3 16823* The values of the converse/inverse of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremisclo 16824* A set is clopen iff for every point in the space there is a neighborhood such that all the points in are in iff is. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theoremisclo2 16825* A set is clopen iff for every point in the space there is a neighborhood of which is either disjoint from or contained in . (Contributed by Mario Carneiro, 7-Jul-2015.)

Theoremdiscld 16826 The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.)

Theoremsn0cld 16827 The closed sets of the topology . (Contributed by FL, 5-Jan-2009.)

Theoremindiscld 16828 The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremmretopd 16829* A Moore collection which is closed under finite unions called topological; such a collection is the closed sets of a canonically associated topology. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Moore                            TopOn

Theoremtoponmre 16830 The topologies over a given base set form a Moore collection: the intersection of any family of them is a topology, including the empty (relative) intersection which gives the discrete topology distop 16733. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 5-May-2015.)
TopOn Moore

Theoremcldmreon 16831 The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn Moore

Theoremiscldtop 16832* A family is the closed sets of a topology iff it is a Moore collection and closed under finite union. (Contributed by Stefan O'Rear, 1-Feb-2015.)
TopOn Moore

Theoremmreclatdemo 16833 The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclat 14290. (Contributed by Stefan O'Rear, 31-Jan-2015.)
toInc

11.1.5  Neighborhoods

Syntaxcnei 16834 Extend class notation with neighborhood relation for topologies.

Definitiondf-nei 16835* Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)

Theoremneifval 16836* 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 16837 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 16838 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 16839* 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 16840* The predicate " is a neighborhood of ." (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremneiint 16841 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 16842* The predicate " is a neighborhood of point ." (Contributed by NM, 26-Feb-2007.)

Theoremneii1 16843 A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)

Theoremneisspw 16844 The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)

Theoremneii2 16845* Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)

Theoremneiss 16846 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 16847 A set is included in its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3 . (Contributed by FL, 16-Nov-2006.)

Theoremelnei 16848 A point belongs to any of its neighborhoods. Proposition Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)

Theorem0nnei 16849 The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)

Theoremneips 16850* 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 16851 An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)

Theoremopnssneib 16852 Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)

Theoremssnei2 16853 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 16854 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 16855 An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)

Theoremopnneip 16856 An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)

Theoremopnnei 16857* 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 16858 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 16855. (Contributed by FL, 2-Oct-2006.)

Theoremneiuni 16859 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 16860* A point belongs to the closure of a set iff every neighborhood of meets . (Contributed by FL, 15-Sep-2013.)

Theoremtopssnei 16861 A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)

Theoreminnei 16862 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 16863 Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)

Theoremneissex 16864* 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 16865 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)

11.1.6  Limit points and perfect sets

Syntaxclp 16866 Extend class notation with the limit point function for topologies.

Syntaxcperf 16867 Extend class notation with the class of all perfect spaces.
Perf

Definitiondf-lp 16868* Define a function on topologies whose value is the set of limit points of the subsets of the base set. See lpval 16871. (Contributed by NM, 10-Feb-2007.)

Definitiondf-perf 16869 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 16870* 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 16871* 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 16872 The predicate " is a limit point of ." (Contributed by NM, 10-Feb-2007.)

Theoremlpsscls 16873 The limits points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007.)

Theoremlpss 16874 The limits points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007.)

Theoremlpdifsn 16875 is a limit point of iff it is a limit point of . (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremlpss3 16876 Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremislp2 16877* 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 16878 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 16879 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 16880 A point belonging to a set's closure but not the set itself is a limit point. (Contributed by NM, 8-Nov-2007.)

Theoremcldlp 16881 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 16882 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremisperf2 16883 Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremisperf3 16884* A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperflp 16885 The limit points of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperfi 16886 Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Perf

Theoremperftop 16887 A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)
Perf

11.1.7  Subspace topologies

Theoremrestrcl 16888 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
t

Theoremrestbas 16889 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 16890 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 16891 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 16892 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrestuni 16893 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
t

Theoremstoig 16894 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 16895 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
t t t

Theoremrestabs 16896 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 16897 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 16898 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theoremresttopon2 16899 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrest0 16900 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

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