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Theorem List for Intuitionistic Logic Explorer - 12001-12100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcls0 12001 The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)

Theoremntr0 12002 The interior of the empty set. (Contributed by NM, 2-Oct-2007.)

Theoremisopn3i 12003 An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremdiscld 12004 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 12005 The closed sets of the topology . (Contributed by FL, 5-Jan-2009.)

6.1.5  Neighborhoods

Syntaxcnei 12006 Extend class notation with neighborhood relation for topologies.

Definitiondf-nei 12007* Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)

Theoremneifval 12008* Value of the neighborhood function on the subsets of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremneif 12009 The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremneiss2 12010 A set with a neighborhood is a subset of the base set of a topology. (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 12011* Value of 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 12012* The predicate "the class is a neighborhood of ". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremneiint 12013 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 12014* The predicate "the class is a neighborhood of point ". (Contributed by NM, 26-Feb-2007.)

Theoremneii1 12015 A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)

Theoremneisspw 12016 The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)

Theoremneii2 12017* Property of a neighborhood. (Contributed by NM, 12-Feb-2007.)

Theoremneiss 12018 Any neighborhood of a set is also a neighborhood of any subset . Similar to Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)

Theoremssnei 12019 A set is included in any of its neighborhoods. Generalization to subsets of elnei 12020. (Contributed by FL, 16-Nov-2006.)

Theoremelnei 12020 A point belongs to any of its neighborhoods. Property Viii of [BourbakiTop1] p. I.3. (Contributed by FL, 28-Sep-2006.)

Theorem0nnei 12021 The empty set is not a neighborhood of a nonempty set. (Contributed by FL, 18-Sep-2007.)

Theoremneipsm 12022* A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)

Theoremopnneissb 12023 An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)

Theoremopnssneib 12024 Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)

Theoremssnei2 12025 Any subset of containing a neighborhood of a set is a neighborhood of this set. Generalization to subsets of Property Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)

Theoremopnneiss 12026 An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.)

Theoremopnneip 12027 An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.)

Theoremtpnei 12028 The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 12026. (Contributed by FL, 2-Oct-2006.)

Theoremneiuni 12029 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.)

Theoremtopssnei 12030 A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)

Theoreminnei 12031 The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.)

Theoremopnneiid 12032 Only an open set is a neighborhood of itself. (Contributed by FL, 2-Oct-2006.)

Theoremneissex 12033* For any neighborhood of , there is a neighborhood of such that is a neighborhood of all subsets of . Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)

Theorem0nei 12034 The empty set is a neighborhood of itself. (Contributed by FL, 10-Dec-2006.)

6.1.6  Subspace topologies

Theoremrestrcl 12035 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
t

Theoremrestbasg 12036 A subspace topology basis is a basis. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

Theoremtgrest 12037 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 12038 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 12039 A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrestuni 12040 The underlying set of a subspace topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
t

Theoremstoig 12041 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 12042 Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
t t t

Theoremrestabs 12043 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 12044 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 12045 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
t

Theoremresttopon2 12046 The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn t TopOn

Theoremrest0 12047 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 12048 The only subspace topology induced by the topology . (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
t

Theoremrestopnb 12049 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 12050 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 12051 If is open, then is open in iff it is an open subset of . (Contributed by Mario Carneiro, 2-Mar-2015.)
t

Theoremrestdis 12052 A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
t

6.1.7  Limits and continuity in topological spaces

Syntaxccn 12053 Extend class notation with the class of continuous functions between topologies.

Syntaxccnp 12054 Extend class notation with the class of functions between topologies continuous at a given point.

Syntaxclm 12055 Extend class notation with a function on topological spaces whose value is the convergence relation for limit sequences in the space.

Definitiondf-cn 12056* 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 12064 for the predicate form. (Contributed by NM, 17-Oct-2006.)

Definitiondf-cnp 12057* 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 12058* Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although 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 converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.)

Theoremlmrcl 12059 Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)

Theoremlmfval 12060* The relation "sequence converges to point " in a metric space. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremlmreltop 12061 The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)

Theoremcnfval 12062* 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 12063* 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 12064* The predicate "the class 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 12065* 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 12066* The predicate "the class 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 12067* The predicate "the class is a continuous function from topology to topology ". Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop1 12068 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremcntop2 12069 Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremiscnp3 12070* The predicate "the class is a continuous function from topology to topology at point ". (Contributed by NM, 15-May-2007.)
TopOn TopOn

Theoremcnf 12071 A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnf2 12072 A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn TopOn

Theoremcnprcl2k 12073 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
TopOn

Theoremcnpf2 12074 A continuous function at point is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
TopOn TopOn

Theoremtgcn 12075* 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.)
TopOn              TopOn

Theoremtgcnp 12076* 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.)
TopOn              TopOn

Theoremssidcn 12077 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.)
TopOn TopOn

Theoremicnpimaex 12078* Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
TopOn TopOn

Theoremidcn 12079 A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
TopOn

Theoremlmbr 12080* Express the binary relation "sequence converges to point " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 12058. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbr2 12081* Express the binary relation "sequence converges to point " in a metric space using an arbitrary upper set of integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmbrf 12082* Express the binary relation "sequence converges to point " in a metric space using an arbitrary upper set of integers. This version of lmbr2 12081 presupposes that is a function. (Contributed by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmconst 12083 A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
TopOn

Theoremlmcvg 12084* Convergence property of a converging sequence. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremiscnp4 12085* The predicate "the class is a continuous function from topology to topology at point " in terms of neighborhoods. (Contributed by FL, 18-Jul-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
TopOn TopOn

Theoremcnpnei 12086* A condition for continuity at a point in terms of neighborhoods. (Contributed by Jeff Hankins, 7-Sep-2009.)

Theoremcnima 12087 An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.)

Theoremcnco 12088 The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Theoremcnptopco 12089 The composition of a function continuous at with a function continuous at is continuous at . Proposition 2 of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremcnclima 12090 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.)

Theoremcnntri 12091 Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)

Theoremcnntr 12092* Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
TopOn TopOn

Theoremcnss1 12093 If the topology is finer than , then there are more continuous functions from than from . (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcnss2 12094 If the topology is finer than , then there are fewer continuous functions into than into from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremcncnpi 12095 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.)

Theoremcnsscnp 12096 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.)

Theoremcncnp 12097* 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.)
TopOn TopOn

Theoremcncnp2m 12098* A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.)

Theoremcnnei 12099* Continuity in terms of neighborhoods. (Contributed by Thierry Arnoux, 3-Jan-2018.)

Theoremcnconst2 12100 A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
TopOn TopOn

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