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PART 7  BASIC TOPOLOGY

7.1  Topology

7.1.1  Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.

7.1.1.1  Topologies

Syntaxctop 12201 Syntax for the class of topologies.

Definitiondf-top 12202* Define the class of topologies. It is a proper class. See istopg 12203 and istopfin 12204 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

Theoremistopg 12203* Express the predicate " is a topology". See istopfin 12204 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremistopfin 12204* Express the predicate " is a topology" using nonempty finite intersections instead of binary intersections as in istopg 12203. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)

Theoremuniopn 12205 The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)

Theoremiunopn 12206* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)

Theoreminopn 12207 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

Theoremfiinopn 12208 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)

Theoremunopn 12209 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorem0opn 12210 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)

Theorem0ntop 12211 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)

Theoremtopopn 12212 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

Theoremeltopss 12213 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)

7.1.1.2  Topologies on sets

Syntaxctopon 12214 Syntax for the function of topologies on sets.
TopOn

Definitiondf-topon 12215* Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn

Theoremfuntopon 12216 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
TopOn

Theoremistopon 12217 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtopontop 12218 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponuni 12219 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtopontopi 12220 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponunii 12221 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoptopon 12222 Alternative definition of in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoptopon2 12223 A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
TopOn

Theoremtopontopon 12224 A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
TopOn TopOn

Theoremtoponrestid 12225 Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
TopOn       t

Theoremtoponsspwpwg 12226 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
TopOn

Theoremdmtopon 12227 The domain of TopOn is . (Contributed by BJ, 29-Apr-2021.)
TopOn

Theoremfntopon 12228 The class TopOn is a function with domain . (Contributed by BJ, 29-Apr-2021.)
TopOn

Theoremtoponmax 12229 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponss 12230 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremtoponcom 12231 If is a topology on the base set of topology , then is a topology on the base of . (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremtoponcomb 12232 Biconditional form of toponcom 12231. (Contributed by BJ, 5-Dec-2021.)
TopOn TopOn

Theoremtopgele 12233 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn

7.1.1.3  Topological spaces

Syntaxctps 12234 Syntax for the class of topological spaces.

Definitiondf-topsp 12235 Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopOn

Theoremistps 12236 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremistps2 12237 Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)

Theoremtpsuni 12238 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)

Theoremtpstop 12239 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)

Theoremtpspropd 12240 A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtopontopn 12241 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet       TopOn

Theoremtsettps 12242 If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet       TopOn

Theoremistpsi 12243 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)

Theoremeltpsg 12244 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet        TopOn

Theoremeltpsi 12245 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopSet

7.1.2  Topological bases

Syntaxctb 12246 Syntax for the class of topological bases.

Definitiondf-bases 12247* Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 12249). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)

Theoremisbasisg 12248* Express the predicate "the set is a basis for a topology". (Contributed by NM, 17-Jul-2006.)

Theoremisbasis2g 12249* Express the predicate "the set is a basis for a topology". (Contributed by NM, 17-Jul-2006.)

Theoremisbasis3g 12250* Express the predicate "the set is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)

Theorembasis1 12251 Property of a basis. (Contributed by NM, 16-Jul-2006.)

Theorembasis2 12252* Property of a basis. (Contributed by NM, 17-Jul-2006.)

Theoremfiinbas 12253* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorembaspartn 12254* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremtgval 12255* The topology generated by a basis. See also tgval2 12257 and tgval3 12264. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremtgvalex 12256 The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)

Theoremtgval2 12257* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 12270) that is indeed a topology (on , see unitg 12268). See also tgval 12255 and tgval3 12264. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg 12258 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg2 12259* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg2b 12260* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremeltg4i 12261 An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremeltg3i 12262 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremeltg3 12263* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)

Theoremtgval3 12264* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 12255 and tgval2 12257. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremtg1 12265 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)

Theoremtg2 12266* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)

Theorembastg 12267 A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Theoremunitg 12268 The topology generated by a basis is a topology on . Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)

Theoremtgss 12269 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)

Theoremtgcl 12270 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)

Theoremtgclb 12271 The property tgcl 12270 can be reversed: if the topology generated by is actually a topology, then must be a topological basis. This yields an alternative definition of . (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtgtopon 12272 A basis generates a topology on . (Contributed by Mario Carneiro, 14-Aug-2015.)
TopOn

Theoremtopbas 12273 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)

Theoremtgtop 12274 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)

Theoremeltop 12275 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)

Theoremeltop2 12276* Membership in a topology. (Contributed by NM, 19-Jul-2006.)

Theoremeltop3 12277* Membership in a topology. (Contributed by NM, 19-Jul-2006.)

Theoremtgdom 12278 A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)

Theoremtgiun 12279* The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtgidm 12280 The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)

Theorembastop 12281 Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)

Theoremtgtop11 12282 The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)

Theoremen1top 12283 is the only topology with one element. (Contributed by FL, 18-Aug-2008.)

Theoremtgss3 12284 A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theoremtgss2 12285* A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theorembasgen 12286 Given a topology , show that a subset satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)

Theorembasgen2 12287* Given a topology , show that a subset satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theorem2basgeng 12288 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)

Theorembastop1 12289* A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " " to express " is a basis for topology " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Theorembastop2 12290* A version of bastop1 12289 that doesn't have in the antecedent. (Contributed by NM, 3-Feb-2008.)

7.1.3  Examples of topologies

Theoremdistop 12291 The discrete topology on a set . Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)

Theoremtopnex 12292 The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4377. (Contributed by BJ, 2-May-2021.)

Theoremdistopon 12293 The discrete topology on a set , with base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremsn0topon 12294 The singleton of the empty set is a topology on the empty set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremsn0top 12295 The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.) (Proof shortened by Mario Carneiro, 13-Aug-2015.)

Theoremepttop 12296* The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
TopOn

Theoremdistps 12297 The discrete topology on a set expressed as a topological space. (Contributed by FL, 20-Aug-2006.)
TopSet

7.1.4  Closure and interior

Syntaxccld 12298 Extend class notation with the set of closed sets of a topology.

Syntaxcnt 12299 Extend class notation with interior of a subset of a topology base set.

Syntaxccl 12300 Extend class notation with closure of a subset of a topology base set.

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