View source for List of topological properties

{{Stub page|grade=A*|msg=Needs linking in to places. Because density is SPRAWLED all over the place right now}}
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==Index==
Here {{M|(X,\mathcal{J})}} is a [[topological space]] or {{M|(X,d)}} is a [[metric space]] in the definitions.
{| class="wikitable" border="1" style="width:100%;"
|-
! Property
! Topological version
! Metric spaces version
! Comments
|-
! {{link|Closure|topology}}
| Let {{M|A\in\mathcal{P}(X)}} be given. The ''closure'' of {{M|A}}, denoted {{M|\overline{A} }} is defined as follows:
* {{M|\overline{A}:\eq\bigcap\left\{C\in\mathcal{C}(X)\ \big\vert\ A\subseteq C\right\} }}{{rFAVIDMH}} - where {{M|\mathcal{C}(X)}} denotes the set of {{plural|closed set|s}} of {{M|X}}
Informally, it is the smallest [[closed set]] containing {{M|A}}. 
* Note that the largest closed set c
| Probably something with limit points
| See also:
* {{link|Interior|topology}}
* {{link|Boundary|topology}}
|-
! rowspan="3" | [[Dense set]]
| For {{M|A\in\mathcal{P}(X)}} we say {{M|A}} is ''dense'' in {{M|X}} if:
* {{M|\forall U\in\mathcal{J}[U\cap A\neq\emptyset]}}{{rFAVIDMH}}<ref group="Note">There are a few simple equivalent conditions, any of these may be the definition given in a book, although [[closure (topology)|{{M|\text{Closure}(A)}}]]{{M|\eq X}} is quite common</ref>
| For {{M|A\in\mathcal{P}(X)}} we say {{M|A}} is ''dense'' in {{M|X}} if:
* {{M|\forall x\in X\forall\epsilon>0[B_\epsilon(x)\cap A\neq\emptyset]}}{{rFAVIDMH}}
{{Caveat|This is given as equiv to density by{{rFAVIDMH}} - also obviously follows from it!}}
| rowspan="3" | See also:
* [[nowhere dense set]]
|-
! colspan="2" | Equivalent statements
|-
| The following are equivalent to the definition above.
# [[Closure (topology)|{{M|\text{Closure}(A)}}]]{{M|\eq X}}{{rFAVIDMH}}
# {{M|X-A}} contains no (non-empty) [[open set|open subsets]] of {{M|X}}{{rFAVIDMH}}
#* 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)
#* 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}}
| 
|-
! {{link|Interior|topology}}
| {{MM|\text{Int}(A,X):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} }U}}{{rITTMJML}}
|
| Could be union of all interior points, see [https://wiki.unifiedmathematics.com/index.php?title=Interior&oldid=1412 here]
|-
! rowspan="1" | {{link|Interior point|topology}}
| For a set {{M|A\in\mathcal{P}(X)}} and {{M|a\in A}}, {{M|a}} is an ''interior point'' of {{M|A}} if:
* {{M|\exists U\in\mathcal{J}[a\in U\wedge U\subseteq A]}}{{rFAVIDMH}}
| For a set {{M|A\in\mathcal{P}(X)}} and {{M|a\in A}}, {{M|a}} is an ''interior point'' of {{M|A}} if:
* {{M|\exists\epsilon>0[B_\epsilon(a)\subseteq A]}}{{rFAVIDMH}}
{{Caveat|Basically follows from topological definition, these are closely related}}
| rowspan="1" | 
|-
|}

==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Topology|Metric Space|Functional Analysis}}