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)
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#* Symbolically.... {{XXX|Can't be bothered right now, include later}}
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#* 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]}}
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#*: {{M|\iff\forall p\in X-A\forall U\in\mathcal{J}[\neg(p\in U\wedge U\subseteq A)]}}
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#*: {{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]]
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#*: {{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:
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#*: {{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]}}
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{{XXX|Tidy this up}}
 
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==Notes==
 
==Notes==
 
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<references group="Note"/>

Revision as of 04:12, 1 January 2017

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Needs linking in to places. Because density is SPRAWLED all over the place right now

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
Dense set For [ilmath]A\in\mathcal{P}(X)[/ilmath] we say [ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if:
  • [ilmath]\forall U\in\mathcal{J}[U\cap A\neq\emptyset][/ilmath][1][Note 1]
 For [ilmath]A\in\mathcal{P}(X)[/ilmath] we say [ilmath]A[/ilmath] is dense in [ilmath]X[/ilmath] if:
  • [ilmath]\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap A\neq\emptyset][/ilmath][1]

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.
  1. [ilmath]\text{Closure}(A)[/ilmath][ilmath]\eq X[/ilmath][1]
  2. [ilmath]X-A[/ilmath] contains no (non-empty) open subsets of [ilmath]X[/ilmath][1]
    • Symbolically: [ilmath]\forall U\in\mathcal{J}[U\nsubseteq X-A][/ilmath], which we can easily manipulate to get: [ilmath]\forall U\in\mathcal{J}\exists p\in U[p\notin X-M][/ilmath]
  3. [ilmath]X-A[/ilmath] has no interior points[1] (see below)
    • Symbolically we may write this as: [ilmath]\forall p\in X-A\left[\neg\left(\exists U\in\mathcal{J}[p\in U\wedge U\subseteq A\right)\right][/ilmath]
      [ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[\neg(p\in U\wedge U\subseteq A)][/ilmath]
      [ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[(\neg(p\in U))\vee(\neg(U\subseteq A))][/ilmath] - by the negation of logical and
      [ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}[p\notin U\vee U\nsubseteq A][/ilmath] - of course by the implies-subset relation we see [ilmath](A\subseteq B)\iff(\forall a\in A[a\in B])[/ilmath], thus:
      [ilmath]\iff\forall p\in X-A\forall U\in\mathcal{J}\big[p\notin U\vee(\exists q\in U[q\notin A])\big][/ilmath]
TODO: Tidy this up
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:
  • [ilmath]\exists U\in\mathcal{J}[a\in U\wedge U\subseteq A][/ilmath][1]
 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:
  • [ilmath]\exists\epsilon>0[B_\epsilon(a)\subseteq A][/ilmath][1]

Caveat:Basically follows from topological definition, these are closely related

Notes

  1. ↑ 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

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
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