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Type | Label | Description |
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Statement | ||
Theorem | 2strstr1g 12101 | A constructed two-slot structure. Version of 2strstrg 12098 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.) |
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Theorem | 2strbas1g 12102 | The base set of a constructed two-slot structure. Version of 2strbasg 12099 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.) |
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Theorem | 2strop1g 12103 | The other slot of a constructed two-slot structure. Version of 2stropg 12100 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.) |
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Theorem | basendxnplusgndx 12104 | The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.) |
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Theorem | grpstrg 12105 |
A constructed group is a structure on ![]() ![]() ![]() |
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Theorem | grpbaseg 12106 | The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
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Theorem | grpplusgg 12107 | The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
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Theorem | mulrndx 12108 | Index value of the df-mulr 12074 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | mulrid 12109 | Utility theorem: index-independent form of df-mulr 12074. (Contributed by Mario Carneiro, 8-Jun-2013.) |
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Theorem | mulrslid 12110 |
Slot property of ![]() |
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Theorem | plusgndxnmulrndx 12111 | The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
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Theorem | basendxnmulrndx 12112 | The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
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Theorem | rngstrg 12113 | A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
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Theorem | rngbaseg 12114 | The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
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Theorem | rngplusgg 12115 | The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
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Theorem | rngmulrg 12116 | The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) |
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Theorem | starvndx 12117 | Index value of the df-starv 12075 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | starvid 12118 | Utility theorem: index-independent form of df-starv 12075. (Contributed by Mario Carneiro, 6-Oct-2013.) |
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Theorem | starvslid 12119 |
Slot property of ![]() ![]() |
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Theorem | srngstrd 12120 | A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
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Theorem | srngbased 12121 | The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
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Theorem | srngplusgd 12122 | The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.) |
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Theorem | srngmulrd 12123 | The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) |
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Theorem | srnginvld 12124 | The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) |
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Theorem | scandx 12125 | Index value of the df-sca 12076 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | scaid 12126 | Utility theorem: index-independent form of scalar df-sca 12076. (Contributed by Mario Carneiro, 19-Jun-2014.) |
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Theorem | scaslid 12127 | Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.) |
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Theorem | vscandx 12128 | Index value of the df-vsca 12077 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | vscaid 12129 | Utility theorem: index-independent form of scalar product df-vsca 12077. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
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Theorem | vscaslid 12130 |
Slot property of ![]() |
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Theorem | lmodstrd 12131 | A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
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Theorem | lmodbased 12132 | The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.) |
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Theorem | lmodplusgd 12133 | The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.) |
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Theorem | lmodscad 12134 | The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.) |
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Theorem | lmodvscad 12135 | The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.) |
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Theorem | ipndx 12136 | Index value of the df-ip 12078 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | ipid 12137 | Utility theorem: index-independent form of df-ip 12078. (Contributed by Mario Carneiro, 6-Oct-2013.) |
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Theorem | ipslid 12138 |
Slot property of ![]() |
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Theorem | ipsstrd 12139 | A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
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Theorem | ipsbased 12140 | The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
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Theorem | ipsaddgd 12141 | The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
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Theorem | ipsmulrd 12142 | The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
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Theorem | ipsscad 12143 | The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.) |
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Theorem | ipsvscad 12144 | The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.) |
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Theorem | ipsipd 12145 | The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.) |
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Theorem | tsetndx 12146 | Index value of the df-tset 12079 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | tsetid 12147 | Utility theorem: index-independent form of df-tset 12079. (Contributed by NM, 20-Oct-2012.) |
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Theorem | tsetslid 12148 | Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.) |
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Theorem | topgrpstrd 12149 | A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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Theorem | topgrpbasd 12150 | The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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Theorem | topgrpplusgd 12151 | The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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Theorem | topgrptsetd 12152 | The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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Theorem | plendx 12153 | Index value of the df-ple 12080 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) |
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Theorem | pleid 12154 | Utility theorem: self-referencing, index-independent form of df-ple 12080. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.) |
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Theorem | pleslid 12155 |
Slot property of ![]() |
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Theorem | dsndx 12156 | Index value of the df-ds 12082 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
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Theorem | dsid 12157 | Utility theorem: index-independent form of df-ds 12082. (Contributed by Mario Carneiro, 23-Dec-2013.) |
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Theorem | dsslid 12158 |
Slot property of ![]() |
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Syntax | crest 12159 | Extend class notation with the function returning a subspace topology. |
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Syntax | ctopn 12160 | Extend class notation with the topology extractor function. |
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Definition | df-rest 12161* |
Function returning the subspace topology induced by the topology ![]() ![]() |
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Definition | df-topn 12162 | Define the topology extractor function. This differs from df-tset 12079 when a structure has been restricted using df-ress 12006; 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.) |
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Theorem | restfn 12163 | The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.) |
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Theorem | topnfn 12164 | The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | restval 12165* |
The subspace topology induced by the topology ![]() ![]() |
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Theorem | elrest 12166* | The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
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Theorem | elrestr 12167 | Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
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Theorem | restid2 12168 | The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | restsspw 12169 | The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
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Theorem | restid 12170 | 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.) |
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Theorem | topnvalg 12171 | Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.) |
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Theorem | topnidg 12172 | 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.) |
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Theorem | topnpropgd 12173 | 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.) |
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Syntax | ctg 12174 | Extend class notation with a function that converts a basis to its corresponding topology. |
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Syntax | cpt 12175 | Extend class notation with a function whose value is a product topology. |
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Syntax | c0g 12176 | Extend class notation with group identity element. |
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Syntax | cgsu 12177 | Extend class notation to include finitely supported group sums. |
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Definition | df-0g 12178* |
Define group identity element. Remark: this definition is required here
because the symbol ![]() |
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Definition | df-gsum 12179* |
Define the group sum for the structure ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
1. If
2. If
3. If
4. If |
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Definition | df-topgen 12180* | 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.) |
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Definition | df-pt 12181* | 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.) |
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Syntax | cprds 12182 | The function constructing structure products. |
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Syntax | cpws 12183 | The function constructing structure powers. |
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Definition | df-prds 12184* | 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.) |
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Theorem | reldmprds 12185 | The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) |
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Definition | df-pws 12186* | Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.) |
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Syntax | cpsmet 12187 | Extend class notation with the class of all pseudometric spaces. |
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Syntax | cxmet 12188 | Extend class notation with the class of all extended metric spaces. |
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Syntax | cmet 12189 | Extend class notation with the class of all metrics. |
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Syntax | cbl 12190 | Extend class notation with the metric space ball function. |
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Syntax | cfbas 12191 | Extend class definition to include the class of filter bases. |
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Syntax | cfg 12192 | Extend class definition to include the filter generating function. |
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Syntax | cmopn 12193 | Extend class notation with a function mapping each metric space to the family of its open sets. |
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Syntax | cmetu 12194 | Extend class notation with the function mapping metrics to the uniform structure generated by that metric. |
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Definition | df-psmet 12195* | 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.) |
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Definition | df-xmet 12196* |
Define the set of all extended metrics on a given base set. The
definition is similar to df-met 12197, but we also allow the metric to
take
on the value ![]() |
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Definition | df-met 12197* | 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.) |
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Definition | df-bl 12198* | Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
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Definition | df-mopn 12199 | Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.) |
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Definition | df-fbas 12200* | 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.) |
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