What is Hausdorff space? — Mathematics

Definition 6.7 University Mathematics — Year 3 · Chapter 6 — General Topology

$X$ is Hausdorff (or separated) if any two distinct points have disjoint neighborhoods. Metric spaces are Hausdorff (balls of radius $d(x,y)/2$ ). A sequence $(x_n)$ converges to $x$ if every neighborhood of $x$ contains all but finitely many $x_n$ ; in a Hausdorff space, limits are unique (two limits would have disjoint neighborhoods each containing a tail). In a Hausdorff space, points — hence finite sets — are closed.

Examples

Example 6.11

The quotient $\R/\Z$ (identify $x$ and $x + n$ ) is homeomorphic to the circle $S^1 = \{z \in \C : \abs z = 1\}$ : the map $x \mapsto \eu^{2\iu\pi x}$ passes to a continuous bijection $\R/\Z \to S^1$ (Proposition 6.10(b)); its inverse is continuous by the compactness argument of Corollary 6.14 below (Exercise 6.5 details everything, including why $\R/\Z$ is Hausdorff and compact). Likewise $[0,1]$ with endpoints glued is $S^1$ , the square with opposite sides glued is the torus, and gluing is finally a theorem, not a picture.

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Examples

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