Hausdorff space

From hyperspacewiki
Revision as of 03:48, 29 November 2018 by Jflopezfernandez (talk | contribs) (Created page)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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.