Difference between revisions of "List of topological properties"
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(Created page with "{{Stub page|grade=A*|msg=Needs linking in to places. Because density is SPRAWLED all over the place right now}} __TOC__ ==Index== Here {{M|(X,\mathcal{J})}} is a topological...") |
(→Index: Added symbolic form of equiv 3 to density.) |
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#* Symbolically: {{M|\forall U\in\mathcal{J}[U\nsubseteq X-A]}}, which we can easily manipulate to get: {{M|\forall U\in\mathcal{J}\exists p\in U[p\notin X-M]}} | #* Symbolically: {{M|\forall U\in\mathcal{J}[U\nsubseteq X-A]}}, which we can easily manipulate to get: {{M|\forall U\in\mathcal{J}\exists p\in U[p\notin X-M]}} | ||
# {{M|X-A}} has no {{link|interior point|topology|s}}{{rFAVIDMH}} (see below) | # {{M|X-A}} has no {{link|interior point|topology|s}}{{rFAVIDMH}} (see below) | ||
− | #* Symbolically | + | #* Symbolically we may write this as: {{M|\forall p\in X-A\left[\neg\left(\exists U\in\mathcal{J}[p\in U\wedge U\subseteq A\right)\right]}} |
+ | #*: {{M|\iff\forall p\in X-A\forall U\in\mathcal{J}[\neg(p\in U\wedge U\subseteq A)]}} | ||
+ | #*: {{M|\iff\forall p\in X-A\forall U\in\mathcal{J}[(\neg(p\in U))\vee(\neg(U\subseteq A))]}} - by the [[negation of logical and]] | ||
+ | #*: {{M|\iff\forall p\in X-A\forall U\in\mathcal{J}[p\notin U\vee U\nsubseteq A]}} - of course by the [[implies-subset relation]] we see {{M|(A\subseteq B)\iff(\forall a\in A[a\in B])}}, thus: | ||
+ | #*: {{M|\iff\forall p\in X-A\forall U\in\mathcal{J}\big[p\notin U\vee(\exists q\in U[q\notin A])\big]}} | ||
+ | {{XXX|Tidy this up}} | ||
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==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> |
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Needs linking in to places. Because density is SPRAWLED all over the place right now
Contents
Index
Here [ilmath](X,\mathcal{J})[/ilmath] is a topological space or [ilmath](X,d)[/ilmath] is a metric space in the definitions.
Property | Topological version | Metric spaces version | Comments |
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Dense set | For [ilmath]A\in\mathcal{P}(X)[/ilmath] we say [ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if: | For [ilmath]A\in\mathcal{P}(X)[/ilmath] we say [ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if:
Caveat:This is given as equiv to density by[1] - also obviously follows from it! |
See also: |
Equivalent statements | |||
The following are equivalent to the definition above.
TODO: Tidy this up
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Interior point | For a set [ilmath]A\in\mathcal{P}(X)[/ilmath] and [ilmath]a\in A[/ilmath], [ilmath]a[/ilmath] is an interior point of [ilmath]A[/ilmath] if:
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For a set [ilmath]A\in\mathcal{P}(X)[/ilmath] and [ilmath]a\in A[/ilmath], [ilmath]a[/ilmath] is an interior point of [ilmath]A[/ilmath] if:
Caveat:Basically follows from topological definition, these are closely related |
Notes
- ↑ There are a few simple equivalent conditions, any of these may be the definition given in a book, although [ilmath]\text{Closure}(A)[/ilmath][ilmath]\eq X[/ilmath] is quite common