List of topological properties

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


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
Closure Let [ilmath]A\in\mathcal{P}(X)[/ilmath] be given. The closure of [ilmath]A[/ilmath], denoted [ilmath]\overline{A} [/ilmath] is defined as follows:
  • [ilmath]\overline{A}:\eq\bigcap\left\{C\in\mathcal{C}(X)\ \big\vert\ A\subseteq C\right\} [/ilmath][1] - where [ilmath]\mathcal{C}(X)[/ilmath] denotes the set of closed sets of [ilmath]X[/ilmath]

Informally, it is the smallest closed set containing [ilmath]A[/ilmath].

  • Note that the largest closed set c
Probably something with limit points See also:
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 [math]\text{Int}(A,X):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} }U[/math][2] Could be union of all interior points, see here
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


  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


  1. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
  2. Introduction to Topological Manifolds - John M. Lee