Closure of one-point subset implies irreducible: Difference between revisions
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Suppose <math>X</math> is a [[topological space]] and <math>x</math> is a point in <math>X</math>. Let <math>A</math> be the closure of <math>\{ x \}</math> in <math>X</math>. Then, <matH>A</matH> is an [[irreducible space]] with the [[subspace topology]] from <math>X</math>. | Suppose <math>X</math> is a [[topological space]] and <math>x</math> is a point in <math>X</math>. Let <math>A</math> be the closure of <math>\{ x \}</math> in <math>X</math>. Then, <matH>A</matH> is an [[irreducible space]] with the [[subspace topology]] from <math>X</math>. | ||
==Related facts== | |||
A [[sober space]] is a topological space where the only irreducible closed subsets are the closures of one-point subsets. We have [[Hausdorff implies sober]]. | |||
Latest revision as of 20:11, 13 January 2012
Statement
Suppose is a topological space and is a point in . Let be the closure of in . Then, is an irreducible space with the subspace topology from .
Related facts
A sober space is a topological space where the only irreducible closed subsets are the closures of one-point subsets. We have Hausdorff implies sober.