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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | topgrpplusgd 11701 | 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 11702 | 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 11703 | Index value of the df-ple 11630 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.) |
⊢ (le‘ndx) = ;10 | ||
Theorem | pleid 11704 | Utility theorem: self-referencing, index-independent form of df-ple 11630. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.) |
⊢ le = Slot (le‘ndx) | ||
Theorem | pleslid 11705 | Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.) |
⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ) | ||
Syntax | crest 11706 | Extend class notation with the function returning a subspace topology. |
class ↾t | ||
Syntax | ctopn 11707 | Extend class notation with the topology extractor function. |
class TopOpen | ||
Definition | df-rest 11708* | 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 11709 | Define the topology extractor function. This differs from df-tset 11629 when a structure has been restricted using df-ress 11556; 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 11710 | The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.) |
⊢ ↾t Fn (V × V) | ||
Theorem | topnfn 11711 | The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ TopOpen Fn V | ||
Theorem | restval 11712* | 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 11713* | 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 11714 | 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 11715 | 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 11716 | 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 11717 | 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 11718 | 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 11719 | 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 11720 | 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 11721 | Extend class notation with a function that converts a basis to its corresponding topology. |
class topGen | ||
Syntax | cpt 11722 | Extend class notation with a function whose value is a product topology. |
class ∏t | ||
Syntax | c0g 11723 | Extend class notation with group identity element. |
class 0g | ||
Syntax | cgsu 11724 | Extend class notation to include finitely supported group sums. |
class Σg | ||
Definition | df-0g 11725* | Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 11726. 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 11726* |
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 11727* | 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 11728* | 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 11729 | The function constructing structure products. |
class Xs | ||
Syntax | cpws 11730 | The function constructing structure powers. |
class ↑s | ||
Definition | df-prds 11731* | 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 11732 | 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 11733* | 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 11734 | Extend class notation with the class of all pseudometric spaces. |
class PsMet | ||
Syntax | cxmet 11735 | Extend class notation with the class of all extended metric spaces. |
class ∞Met | ||
Syntax | cmet 11736 | Extend class notation with the class of all metrics. |
class Met | ||
Syntax | cbl 11737 | Extend class notation with the metric space ball function. |
class ball | ||
Syntax | cfbas 11738 | Extend class definition to include the class of filter bases. |
class fBas | ||
Syntax | cfg 11739 | Extend class definition to include the filter generating function. |
class filGen | ||
Syntax | cmopn 11740 | Extend class notation with a function mapping each metric space to the family of its open sets. |
class MetOpen | ||
Syntax | cmetu 11741 | Extend class notation with the function mapping metrics to the uniform structure generated by that metric. |
class metUnif | ||
Definition | df-psmet 11742* | 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 11743* | Define the set of all extended metrics on a given base set. The definition is similar to df-met 11744, 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 11744* | 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 11745* | 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 11746 | 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 11747* | 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 11748* | 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 11749* | 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 11750 | Syntax for the class of topologies. |
class Top | ||
Definition | df-top 11751* |
Define the class of topologies. It is a proper class. See istopg 11752 and
istopfin 11753 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 11752* |
Express the predicate "𝐽 is a topology". See istopfin 11753 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 11753* | Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 11752. 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 11754 | 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 11755* | 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 11756 | 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 11757 | 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 11758 | 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 11759 | The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | ||
Theorem | 0ntop 11760 | The empty set is not a topology. (Contributed by FL, 1-Jun-2008.) |
⊢ ¬ ∅ ∈ Top | ||
Theorem | topopn 11761 | The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) | ||
Theorem | eltopss 11762 | 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 11763 | Syntax for the function of topologies on sets. |
class TopOn | ||
Definition | df-topon 11764* | 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 11765 | The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.) |
⊢ Fun TopOn | ||
Theorem | istopon 11766 | 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 11767 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | ||
Theorem | toponuni 11768 | 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 11769 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐽 ∈ Top | ||
Theorem | toponunii 11770 | 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 11771 | Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) | ||
Theorem | toptopon2 11772 | 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 11773 | 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 | toponsspwpwg 11774 | 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 11775 | The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.) |
⊢ dom TopOn = V | ||
Theorem | fntopon 11776 | The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.) |
⊢ TopOn Fn V | ||
Theorem | toponmax 11777 | The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) | ||
Theorem | toponss 11778 | 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 11779 | 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 11780 | Biconditional form of toponcom 11779. (Contributed by BJ, 5-Dec-2021.) |
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽))) | ||
Theorem | topgele 11781 | 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 11782 | Syntax for the class of topological spaces. |
class TopSp | ||
Definition | df-topsp 11783 | 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 11784 | 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 11785 | 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)) | ||
Theorem | tpsuni 11786 | The base set of a topological space. (Contributed by FL, 27-Jun-2014.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽) | ||
Theorem | tpstop 11787 | The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.) |
⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top) | ||
Theorem | tpspropd 11788 | A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) & ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) ⇒ ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) | ||
Theorem | topontopn 11789 | Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopSet‘𝐾) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) | ||
Theorem | tsettps 11790 | 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.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopSet‘𝐾) ⇒ ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) | ||
Theorem | istpsi 11791 | Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.) |
⊢ (Base‘𝐾) = 𝐴 & ⊢ (TopOpen‘𝐾) = 𝐽 & ⊢ 𝐴 = ∪ 𝐽 & ⊢ 𝐽 ∈ Top ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | eltpsg 11792 | Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} ⇒ ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) | ||
Theorem | eltpsi 11793 | 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.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} & ⊢ 𝐴 = ∪ 𝐽 & ⊢ 𝐽 ∈ Top ⇒ ⊢ 𝐾 ∈ TopSp | ||
Syntax | ctb 11794 | Syntax for the class of topological bases. |
class TopBases | ||
Definition | df-bases 11795* | Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 11797). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.) |
⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))} | ||
Theorem | isbasisg 11796* | Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))) | ||
Theorem | isbasis2g 11797* | Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))) | ||
Theorem | isbasis3g 11798* | Express the predicate "the set 𝐵 is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥 ∈ 𝐵 𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝐵∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))) | ||
Theorem | basis1 11799 | Property of a basis. (Contributed by NM, 16-Jul-2006.) |
⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))) | ||
Theorem | basis2 11800* | Property of a basis. (Contributed by NM, 17-Jul-2006.) |
⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))) |
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