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We have a Noetherian topological space $X$. Show that the following are equivalent:

  1. $X$ is a Hausdorff space
  2. $X$ is finite and has discrete topology

So far I've only got this:

If $X$ has discrete topology, then any two points $x_1$ and $x_2$ are elements of the disjoint open sets $\{x_1\}$ and $\{x_2\}$ respectively. This implies $X$ is Hausdorff. (So 2$\implies1$)

I don't know how to prove ($1\implies 2$).

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    $\begingroup$ You want to use (ir)reducibility for this. Proceed by contradiction. $\endgroup$ May 19, 2015 at 17:29
  • $\begingroup$ Note that, for (1)$\Rightarrow$(2) you just need to show that $X$ is finite, because a finite Hausdorff space is necessarily discrete. $\endgroup$
    – egreg
    May 19, 2015 at 17:46
  • $\begingroup$ @egreg Just showing discreteness should also suffice I found, due to the Noetherian property. Is proving finiteness easier? $\endgroup$
    – gebruiker
    May 19, 2015 at 17:50
  • $\begingroup$ I believe I found it. I'm posting a self-answer for clarity. $\endgroup$
    – gebruiker
    May 19, 2015 at 17:52

2 Answers 2

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Assume first that $X$ is irreducible. Then every nonempty open $U$ subset of $X$ is dense. But if $X$ is Hausdorff the existence of a point in the complement of $U$ is impossible. Therefore every nonempty open subset of $X$ is $X$ itself. Again since $X$ is Hausdorff this implies that $X$ consists only of one point.

In general, since $X$ is noetherian you can write it as a finite union of its irreducible components $X_1,\ldots,X_n$. Since $X$ is Hausdorff the $X_i$'s are also Hausdorff and thus by the above consist of one point. Therefore $X$ is finite with $n$ points and every subset consisting of one point is closes. This means that $X$ has the discrete topology.

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$X$ is a Noetherian topological space, so there exist a nonnegative integer $n$ and closed, irreducible subsets $Z_1,\ldots, Z_n$ such that $X=Z_1\cup\ldots\cup Z_n$, with $Z_i\not\subseteq Z_j$ for $i\neq j$.

Now suppose we have two distinct points $x,y\in Z_i$. The Hausdorff property of $X$ implies that we must now also be able to find disjoint open sets $U_1\ni x$ and $U_2\ni y$. In the subspace topology of $Z_i$ we now see that the two open sets $U_1\cap Z_i$ and $U_2\cap Z_i$ are disjoint. This contradicts the fact that $Z_i$ is irreducible. Hence we cannot find two distinct points in any $Z_i$, which in turn implies that $X$ is a finite union of singletons. Hence $X$ is a finite Hausdorff space which necessarily means it also has the discrete topology.

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