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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | topgrpplusgd 12101 | The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
TopSet | ||
Theorem | topgrptsetd 12102 | The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
TopSet TopSet | ||
Theorem | plendx 12103 | Index value of the df-ple 12030 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) |
; | ||
Theorem | pleid 12104 | Utility theorem: self-referencing, index-independent form of df-ple 12030. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.) |
Slot | ||
Theorem | pleslid 12105 | Slot property of . (Contributed by Jim Kingdon, 9-Feb-2023.) |
Slot | ||
Theorem | dsndx 12106 | Index value of the df-ds 12032 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
; | ||
Theorem | dsid 12107 | Utility theorem: index-independent form of df-ds 12032. (Contributed by Mario Carneiro, 23-Dec-2013.) |
Slot | ||
Theorem | dsslid 12108 | Slot property of . (Contributed by Jim Kingdon, 6-May-2023.) |
Slot | ||
Syntax | crest 12109 | Extend class notation with the function returning a subspace topology. |
↾_{t} | ||
Syntax | ctopn 12110 | Extend class notation with the topology extractor function. |
Definition | df-rest 12111* | Function returning the subspace topology induced by the topology and the set . (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.) |
↾_{t} | ||
Definition | df-topn 12112 | Define the topology extractor function. This differs from df-tset 12029 when a structure has been restricted using df-ress 11956; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopSet ↾_{t} | ||
Theorem | restfn 12113 | The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.) |
↾_{t} | ||
Theorem | topnfn 12114 | The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Theorem | restval 12115* | The subspace topology induced by the topology on the set . (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.) |
↾_{t} | ||
Theorem | elrest 12116* | The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
↾_{t} | ||
Theorem | elrestr 12117 | Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
↾_{t} | ||
Theorem | restid2 12118 | The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
↾_{t} | ||
Theorem | restsspw 12119 | The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
↾_{t} | ||
Theorem | restid 12120 | The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.) |
↾_{t} | ||
Theorem | topnvalg 12121 | Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.) |
TopSet ↾_{t} | ||
Theorem | topnidg 12122 | Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopSet | ||
Theorem | topnpropgd 12123 | The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.) |
TopSet TopSet | ||
Syntax | ctg 12124 | Extend class notation with a function that converts a basis to its corresponding topology. |
Syntax | cpt 12125 | Extend class notation with a function whose value is a product topology. |
Syntax | c0g 12126 | Extend class notation with group identity element. |
Syntax | cgsu 12127 | Extend class notation to include finitely supported group sums. |
_{g} | ||
Definition | df-0g 12128* | Define group identity element. Remark: this definition is required here because the symbol is already used in df-gsum 12129. The related theorems will be provided later. (Contributed by NM, 20-Aug-2011.) |
Definition | df-gsum 12129* |
Define the group sum for the structure of a finite sequence of
elements whose values are defined by the expression and whose set
of indices is .
It may be viewed as a product (if is a
multiplication), a sum (if is an addition) or any other operation.
The variable is
normally a free variable in (i.e., can
be thought of as ). The definition is meaningful in
different contexts, depending on the size of the index set and
each demanding different properties of .
1. If and has an identity element, then the sum equals this identity. 2. If and is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e. etc. 3. If is a finite set (or is nonzero for finitely many indices) and is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined. 4. If is an infinite set and is a Hausdorff topological group, then there is a meaningful sum, but _{g} cannot handle this case. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.) |
_{g} ♯ ♯ | ||
Definition | df-topgen 12130* | Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.) |
Definition | df-pt 12131* | Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Syntax | cprds 12132 | The function constructing structure products. |
_{s} | ||
Syntax | cpws 12133 | The function constructing structure powers. |
_{s} | ||
Definition | df-prds 12134* | Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) |
_{s} Scalar _{g} TopSet comp comp | ||
Theorem | reldmprds 12135 | The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) |
_{s} | ||
Definition | df-pws 12136* | Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.) |
_{s} Scalar_{s} | ||
Syntax | cpsmet 12137 | Extend class notation with the class of all pseudometric spaces. |
PsMet | ||
Syntax | cxmet 12138 | Extend class notation with the class of all extended metric spaces. |
Syntax | cmet 12139 | Extend class notation with the class of all metrics. |
Syntax | cbl 12140 | Extend class notation with the metric space ball function. |
Syntax | cfbas 12141 | Extend class definition to include the class of filter bases. |
Syntax | cfg 12142 | Extend class definition to include the filter generating function. |
Syntax | cmopn 12143 | Extend class notation with a function mapping each metric space to the family of its open sets. |
Syntax | cmetu 12144 | Extend class notation with the function mapping metrics to the uniform structure generated by that metric. |
metUnif | ||
Definition | df-psmet 12145* | Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
PsMet | ||
Definition | df-xmet 12146* | Define the set of all extended metrics on a given base set. The definition is similar to df-met 12147, but we also allow the metric to take on the value . (Contributed by Mario Carneiro, 20-Aug-2015.) |
Definition | df-met 12147* | Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. (Contributed by NM, 25-Aug-2006.) |
Definition | df-bl 12148* | Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Definition | df-mopn 12149 | Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.) |
Definition | df-fbas 12150* | Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.) |
Definition | df-fg 12151* | Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.) |
Definition | df-metu 12152* | Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
metUnif PsMet | ||
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. | ||
Syntax | ctop 12153 | Syntax for the class of topologies. |
Definition | df-top 12154* |
Define the class of topologies. It is a proper class. See istopg 12155 and
istopfin 12156 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.) |
Theorem | istopg 12155* |
Express the predicate " is a topology". See istopfin 12156 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.) |
Theorem | istopfin 12156* | Express the predicate " is a topology" using nonempty finite intersections instead of binary intersections as in istopg 12155. 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.) |
Theorem | uniopn 12157 | 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.) |
Theorem | iunopn 12158* | The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.) |
Theorem | inopn 12159 | The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Theorem | fiinopn 12160 | The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.) |
Theorem | unopn 12161 | The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | 0opn 12162 | The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
Theorem | 0ntop 12163 | The empty set is not a topology. (Contributed by FL, 1-Jun-2008.) |
Theorem | topopn 12164 | The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Theorem | eltopss 12165 | A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.) |
Syntax | ctopon 12166 | Syntax for the function of topologies on sets. |
TopOn | ||
Definition | df-topon 12167* | Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
TopOn | ||
Theorem | funtopon 12168 | The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.) |
TopOn | ||
Theorem | istopon 12169 | 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 | ||
Theorem | topontop 12170 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toponuni 12171 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | topontopi 12172 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toponunii 12173 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toptopon 12174 | Alternative definition of in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toptopon2 12175 | A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
TopOn | ||
Theorem | topontopon 12176 | A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
TopOn TopOn | ||
Theorem | toponrestid 12177 | Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.) |
TopOn ↾_{t} | ||
Theorem | toponsspwpwg 12178 | 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 | ||
Theorem | dmtopon 12179 | The domain of TopOn is . (Contributed by BJ, 29-Apr-2021.) |
TopOn | ||
Theorem | fntopon 12180 | The class TopOn is a function with domain . (Contributed by BJ, 29-Apr-2021.) |
TopOn | ||
Theorem | toponmax 12181 | The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | toponss 12182 | A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
TopOn | ||
Theorem | toponcom 12183 | 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 | ||
Theorem | toponcomb 12184 | Biconditional form of toponcom 12183. (Contributed by BJ, 5-Dec-2021.) |
TopOn TopOn | ||
Theorem | topgele 12185 | 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 | ||
Syntax | ctps 12186 | Syntax for the class of topological spaces. |
Definition | df-topsp 12187 | Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.) |
TopOn | ||
Theorem | istps 12188 | Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopOn | ||
Theorem | istps2 12189 | Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.) |
Theorem | tpsuni 12190 | The base set of a topological space. (Contributed by FL, 27-Jun-2014.) |
Theorem | tpstop 12191 | The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.) |
Theorem | tpspropd 12192 | A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Theorem | topontopn 12193 | Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopSet TopOn | ||
Theorem | tsettps 12194 | 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 | ||
Theorem | istpsi 12195 | Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.) |
Theorem | eltpsg 12196 | Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) |
TopSet TopOn | ||
Theorem | eltpsi 12197 | 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 | ||
Syntax | ctb 12198 | Syntax for the class of topological bases. |
Definition | df-bases 12199* | Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 12201). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.) |
Theorem | isbasisg 12200* | Express the predicate "the set is a basis for a topology". (Contributed by NM, 17-Jul-2006.) |
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