Difference between revisions of "Every sequence in a compact space is a lingering sequence/Statement"
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In a [[metric space]] {{M|(X,d)}} that is [[compact]] every [[sequence]] is a [[lingering sequence]], that is to say{{rITTGG}}: | In a [[metric space]] {{M|(X,d)}} that is [[compact]] every [[sequence]] is a [[lingering sequence]], that is to say{{rITTGG}}: | ||
| − | * {{MM|1=\forall(x_n)_{n=1}^\infty\subseteq X\ \exists x\in X\ \forall\epsilon>0[\vert B_\epsilon(x)\cap(x_n)_{n=1}^\infty\vert=\aleph_0]}} | + | * {{MM|1=\forall(x_n)_{n=1}^\infty\subseteq X\ :\ \exists x\in X\ \forall\epsilon>0[\vert B_\epsilon(x)\cap(x_n)_{n=1}^\infty\vert=\aleph_0]}} |
</onlyinclude> | </onlyinclude> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
Latest revision as of 16:22, 6 December 2015
Statement
In a metric space [ilmath](X,d)[/ilmath] that is compact every sequence is a lingering sequence, that is to say[1]:
- [math]\forall(x_n)_{n=1}^\infty\subseteq X\ :\ \exists x\in X\ \forall\epsilon>0[\vert B_\epsilon(x)\cap(x_n)_{n=1}^\infty\vert=\aleph_0][/math]
