Hausdorff space

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The Hausdorff condition is one of several additional conditions one can impose on a topological space. By adding additional constraints, the theorems that may be proven are stronger, but they apply to fewer classes of topological spaces. Hausdorff spaces are also called \$T_2\$ spaces,

A topological space \$S\$ is said to be a Hausdorff space if any two points \$x\$ and \$y\$ can be separated by disjoint neighborhoods \$N_x\$ and \$N_y\$.

Intuitively, this can be explained simply using the topological space comprised of only the real number line \$\mathbb{R}\$. In this space, if the limit of a function \$f(x)\$ exists, it must be unique. There are topological spaces which do not meet this requirement, however, and are thus said to not be closed.[footnotes 1] A Hausdorff space can thus be thought of as a topological space where the "limit" at a point must be unique. In Topology, however, the conventions of limit points and neighborhoods are used, as they make more intuitive sense in dimensions greater than \$\mathbb{R}\$.[1]

Proof

THEOREM: If \$X\$ is a Hausdorff space, then a sequence of points of \$X\$ converges to at most one point of \$X\$.
Proof:
Suppose that \$x_n\$ is a sequence of points of \$X\$ that converges to \$x\$. If \$y \neq x\$, let \$U\$ and \$V\$ be disjoint neighborhoods of \$x\$ and \$y\$, respectively. Since \$U\$ contains \$x_n\$ for all but finitely many values of \$n\$, the set \$V\$ cannot. Therefore, \$x_n\$ cannot converge to \$y\$.[2]

Footnotes

1. It must be noted that the Law of the Excluded Middle does not automatically apply in this case; the fact that a set is not closed does not automatically imply it is open.

References

1. Yandl, André L., and Adam Bowers. Elementary Point-Set Topology: A Transition to Advanced Mathematics. Aurora Dover Modern Math Originals. Mineola, New York: Dover Publications, Inc, 2016.
2. Munkres, James R. Topology, 2015.