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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ipsbased 12101 | The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐴)) | ||
Theorem | ipsaddgd 12102 | The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → + = (+g‘𝐴)) | ||
Theorem | ipsmulrd 12103 | The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → × = (.r‘𝐴)) | ||
Theorem | ipsscad 12104 | 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.) |
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝑆 = (Scalar‘𝐴)) | ||
Theorem | ipsvscad 12105 | 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.) |
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → · = ( ·𝑠 ‘𝐴)) | ||
Theorem | ipsipd 12106 | The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.) |
⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → × ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ (𝜑 → · ∈ 𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐼 = (·𝑖‘𝐴)) | ||
Theorem | tsetndx 12107 | Index value of the df-tset 12040 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (TopSet‘ndx) = 9 | ||
Theorem | tsetid 12108 | Utility theorem: index-independent form of df-tset 12040. (Contributed by NM, 20-Oct-2012.) |
⊢ TopSet = Slot (TopSet‘ndx) | ||
Theorem | tsetslid 12109 | Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.) |
⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | ||
Theorem | topgrpstrd 12110 | A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝑊 Struct 〈1, 9〉) | ||
Theorem | topgrpbasd 12111 | The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝑊)) | ||
Theorem | topgrpplusgd 12112 | The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝑋) ⇒ ⊢ (𝜑 → + = (+g‘𝑊)) | ||
Theorem | topgrptsetd 12113 | The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
⊢ 𝑊 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(TopSet‘ndx), 𝐽〉} & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → + ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐽 = (TopSet‘𝑊)) | ||
Theorem | plendx 12114 | Index value of the df-ple 12041 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ (le‘ndx) = ;10 | ||
Theorem | pleid 12115 | Utility theorem: self-referencing, index-independent form of df-ple 12041. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.) |
⊢ le = Slot (le‘ndx) | ||
Theorem | pleslid 12116 | Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.) |
⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ) | ||
Theorem | dsndx 12117 | Index value of the df-ds 12043 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (dist‘ndx) = ;12 | ||
Theorem | dsid 12118 | Utility theorem: index-independent form of df-ds 12043. (Contributed by Mario Carneiro, 23-Dec-2013.) |
⊢ dist = Slot (dist‘ndx) | ||
Theorem | dsslid 12119 | Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.) |
⊢ (dist = Slot (dist‘ndx) ∧ (dist‘ndx) ∈ ℕ) | ||
Syntax | crest 12120 | Extend class notation with the function returning a subspace topology. |
class ↾t | ||
Syntax | ctopn 12121 | Extend class notation with the topology extractor function. |
class TopOpen | ||
Definition | df-rest 12122* | Function returning the subspace topology induced by the topology 𝑦 and the set 𝑥. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.) |
⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) | ||
Definition | df-topn 12123 | Define the topology extractor function. This differs from df-tset 12040 when a structure has been restricted using df-ress 11967; 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.) |
⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | ||
Theorem | restfn 12124 | The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.) |
⊢ ↾t Fn (V × V) | ||
Theorem | topnfn 12125 | The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ TopOpen Fn V | ||
Theorem | restval 12126* | The subspace topology induced by the topology 𝐽 on the set 𝐴. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.) |
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) | ||
Theorem | elrest 12127* | The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵))) | ||
Theorem | elrestr 12128 | Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) | ||
Theorem | restid2 12129 | The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝐽 ↾t 𝐴) = 𝐽) | ||
Theorem | restsspw 12130 | The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 | ||
Theorem | restid 12131 | 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.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽) | ||
Theorem | topnvalg 12132 | Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopSet‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) | ||
Theorem | topnidg 12133 | 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.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopSet‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐵) → 𝐽 = (TopOpen‘𝑊)) | ||
Theorem | topnpropgd 12134 | 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.) |
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) & ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿)) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑊) ⇒ ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) | ||
Syntax | ctg 12135 | Extend class notation with a function that converts a basis to its corresponding topology. |
class topGen | ||
Syntax | cpt 12136 | Extend class notation with a function whose value is a product topology. |
class ∏t | ||
Syntax | c0g 12137 | Extend class notation with group identity element. |
class 0g | ||
Syntax | cgsu 12138 | Extend class notation to include finitely supported group sums. |
class Σg | ||
Definition | df-0g 12139* | Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 12140. The related theorems will be provided later. (Contributed by NM, 20-Aug-2011.) |
⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)))) | ||
Definition | df-gsum 12140* |
Define the group sum for the structure 𝐺 of a finite sequence of
elements whose values are defined by the expression 𝐵 and
whose set
of indices is 𝐴. It may be viewed as a product (if
𝐺
is a
multiplication), a sum (if 𝐺 is an addition) or any other
operation.
The variable 𝑘 is normally a free variable in 𝐵 (i.e.,
𝐵
can
be thought of as 𝐵(𝑘)). The definition is meaningful in
different contexts, depending on the size of the index set 𝐴 and
each demanding different properties of 𝐺.
1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity. 2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e. (𝐵(1) + 𝐵(2)) + 𝐵(3) etc. 3. If 𝐴 is a finite set (or is nonzero for finitely many indices) and 𝐺 is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined. 4. If 𝐴 is an infinite set and 𝐺 is a Hausdorff topological group, then there is a meaningful sum, but Σg cannot handle this case. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.) |
⊢ Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋{𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g‘𝑤)𝑦) = 𝑦 ∧ (𝑦(+g‘𝑤)𝑥) = 𝑦)} / 𝑜⦌if(ran 𝑓 ⊆ 𝑜, (0g‘𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))), (℩𝑥∃𝑔[(◡𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto→𝑦 ∧ 𝑥 = (seq1((+g‘𝑤), (𝑓 ∘ 𝑔))‘(♯‘𝑦))))))) | ||
Definition | df-topgen 12141* | 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.) |
⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) | ||
Definition | df-pt 12142* | 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.) |
⊢ ∏t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))})) | ||
Syntax | cprds 12143 | The function constructing structure products. |
class Xs | ||
Syntax | cpws 12144 | The function constructing structure powers. |
class ↑s | ||
Definition | df-prds 12145* | 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.) |
⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx), 𝑣〉, 〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx), 𝑠〉, 〈( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}))) | ||
Theorem | reldmprds 12146 | The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) |
⊢ Rel dom Xs | ||
Definition | df-pws 12147* | 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 = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) | ||
Syntax | cpsmet 12148 | Extend class notation with the class of all pseudometric spaces. |
class PsMet | ||
Syntax | cxmet 12149 | Extend class notation with the class of all extended metric spaces. |
class ∞Met | ||
Syntax | cmet 12150 | Extend class notation with the class of all metrics. |
class Met | ||
Syntax | cbl 12151 | Extend class notation with the metric space ball function. |
class ball | ||
Syntax | cfbas 12152 | Extend class definition to include the class of filter bases. |
class fBas | ||
Syntax | cfg 12153 | Extend class definition to include the filter generating function. |
class filGen | ||
Syntax | cmopn 12154 | Extend class notation with a function mapping each metric space to the family of its open sets. |
class MetOpen | ||
Syntax | cmetu 12155 | Extend class notation with the function mapping metrics to the uniform structure generated by that metric. |
class metUnif | ||
Definition | df-psmet 12156* | 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 = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) | ||
Definition | df-xmet 12157* | Define the set of all extended metrics on a given base set. The definition is similar to df-met 12158, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) | ||
Definition | df-met 12158* | 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.) |
⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))}) | ||
Definition | df-bl 12159* | Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧})) | ||
Definition | df-mopn 12160 | Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.) |
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | ||
Definition | df-fbas 12161* | 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.) |
⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)}) | ||
Definition | df-fg 12162* | Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.) |
⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅}) | ||
Definition | df-metu 12163* | 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 = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))))) | ||
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 12164 | Syntax for the class of topologies. |
class Top | ||
Definition | df-top 12165* |
Define the class of topologies. It is a proper class. See istopg 12166 and
istopfin 12167 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.) |
⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} | ||
Theorem | istopg 12166* |
Express the predicate "𝐽 is a topology". See istopfin 12167 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 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) | ||
Theorem | istopfin 12167* | Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 12166. 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.) |
⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥((𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐽))) | ||
Theorem | uniopn 12168 | 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.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽) | ||
Theorem | iunopn 12169* | The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.) |
⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) | ||
Theorem | inopn 12170 | The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) | ||
Theorem | fiinopn 12171 | The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.) |
⊢ (𝐽 ∈ Top → ((𝐴 ⊆ 𝐽 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → ∩ 𝐴 ∈ 𝐽)) | ||
Theorem | unopn 12172 | The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽) | ||
Theorem | 0opn 12173 | The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | ||
Theorem | 0ntop 12174 | The empty set is not a topology. (Contributed by FL, 1-Jun-2008.) |
⊢ ¬ ∅ ∈ Top | ||
Theorem | topopn 12175 | The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) | ||
Theorem | eltopss 12176 | A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Syntax | ctopon 12177 | Syntax for the function of topologies on sets. |
class TopOn | ||
Definition | df-topon 12178* | Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗}) | ||
Theorem | funtopon 12179 | The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.) |
⊢ Fun TopOn | ||
Theorem | istopon 12180 | 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.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) | ||
Theorem | topontop 12181 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | ||
Theorem | toponuni 12182 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | ||
Theorem | topontopi 12183 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐽 ∈ Top | ||
Theorem | toponunii 12184 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐵 = ∪ 𝐽 | ||
Theorem | toptopon 12185 | Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) | ||
Theorem | toptopon2 12186 | A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | ||
Theorem | topontopon 12187 | A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘∪ 𝐽)) | ||
Theorem | toponrestid 12188 | Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.) |
⊢ 𝐴 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐴 = (𝐴 ↾t 𝐵) | ||
Theorem | toponsspwpwg 12189 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) | ||
Theorem | dmtopon 12190 | The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.) |
⊢ dom TopOn = V | ||
Theorem | fntopon 12191 | The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.) |
⊢ TopOn Fn V | ||
Theorem | toponmax 12192 | The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) | ||
Theorem | toponss 12193 | A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Theorem | toponcom 12194 | If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾)) | ||
Theorem | toponcomb 12195 | Biconditional form of toponcom 12194. (Contributed by BJ, 5-Dec-2021.) |
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽))) | ||
Theorem | topgele 12196 | 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.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) | ||
Syntax | ctps 12197 | Syntax for the class of topological spaces. |
class TopSp | ||
Definition | df-topsp 12198 | Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.) |
⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | ||
Theorem | istps 12199 | Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) | ||
Theorem | istps2 12200 | Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
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