5
$\begingroup$

This is an exercise from a topological book.

In $T_1$ space, every compact subset must be closed? For any two compact subset, their intersection must be compact?

Thanks for any help:)

  • 0
    @blindman This is a simplest counterexample I've met. Thanks:)2012-07-31

4 Answers 4

3

Claim 1. If $X$ is a compact subset of a Hausdorff space $(Y, \tau)$ then $X$ is closed.

Proof. It sufficies to show that $Y\setminus X$ is open. To show that $Y\setminus X$ is open, let $y \in Y \setminus X$. For each $x \in X$ fix disjoint $U_x, V_x \in \tau$ so that $x \in U_x$ and $y \in V_x$. From the open cover $\{U_x : x \in X\}$ of $X$ extract a finite subcover, say $\{U_{x_1},U_{x_2},\ldots,U_{x_n}\}$. Then $V_{x_1}\cap V_{x_1},\cap\ldots,\cap V_{x_n}$ is a neighborhood of $y$ in $Y$ that does not intersect $X$.

Claim 2. If $X$ is compact subset of a topological space $(Y, \tau)$ and $K\subset X$ is a closed subset then $K$ is compact.

Proof. Suppose that $\{V_\alpha: \alpha\in I\}\subset \tau$ is an arbitrary open cover for $K$. Since $K$ is closed, $Y\setminus K$ is open. Then $X\subset Y$ is covered by $V_\alpha(\alpha\in I)$ and $Y\setminus K$. Since $X$ is compact, it can be covered by a finite number of open subsets $V_{\alpha_1},\ldots, V_{\alpha_n}$ and $Y\setminus K$, and so is $K\subset X$. Hence $K$ is compact.

Claim 3. If $X_1, X_2$ are closed subsets of a topological space $Y$ then $X_1\cap X_2$ is also closed.

Claim 4. If $X_1, X_2$ are compact subsets of a Hausdorff space $Y$ then $X_1\cap X_2$ is also compact.

Counterexample. If $(Y,\tau)$ is not a Hausdorff space then Claim 1. is not valid. Indeed, let $Y=\{a,b\}$ and $\tau=\{\emptyset, X\}$. Note that $Y$ is not a Hausdorff space. Moreover, $\{a\}$ is a compact subset of $Y$ but $\{a\}$ is not closed.

  • 0
    @blindman Although the space is not $T_1$ which not very help for the question, it is very helpful for me. The proof for the claim 1 is very new. I've upvoted you:)2012-07-31
12

Consider the finite complement topology on an infinite set. Is it $T_1$? Which subsets are compact? Which are closed?

  • 0
    @Michael Yes: For any open cover of the space and for any subset, choose any point from this set, then there exists an open set of the cover whose complement is finite. Then it is obvious... Thanks:)2012-07-31
10

A space that gives negative answers to both questions is the segment $[-1,1]$ with doubled origin.

More precisely, take two copies $I_0 = [-1,1]\times \{0\}$ and $I_1 = [-1,1] \times \{1\}$ of that interval and identify $(t,0)$ with $(t,1)$ whenever $t \neq 0$, that is, we consider the quotient space $Q$ of $[-1,1] \times \{0,1\}$ modulo the equivalence relation generated by $(t,0) \sim (t,1)$ whenever $t \neq 0$.

The space $Q$ is $T_1$: a set in $Q$ is closed if and only if its pre-image under the quotient map is closed, and a pre-image of a point consists of one or two points. Since continuous images of compact sets are compact, the images $C_i$ of $I_i = [-1,1] \times \{i\}$ under the quotient map are both compact, but they are open and non-closed (look at the pre-images under the quotient map again). The intersection $C_0 \cap C_1$ is homeomorphic to $[-1,1] \smallsetminus \{0\}$, in particular it is non-compact.

Note: In order to obtain an example you need to intersect two non-closed compact sets, otherwise the claims in @blindman's answer show that the intersection is compact.

  • 0
    Thanks for the link. I will spend some time to read it carefully:)2012-07-31
4

I would like to advertise the following result, and some consequences, just in case it is not so well reported:

Theorem: Let $B,C$ be compact subsets of $X,Y$ respectively and let $\mathcal W$ be a cover of $B \times C$ by sets open in $X \times Y$. Then $B,C$ have open neighbourhoods $U,V$ respectively such that $U \times V$ is covered by a finite number of sets of $\mathcal W$.

This has a number of consequences from "the product of two compact spaces is compact" to "a compact subset of a Hausdorff space is closed". (Two others are listed in my book "Topology and groupoids", p. 86, but I forget where I first learned of the theorem, back in the 1960s!)

  • 0
    These corollaries are obviously right and given by the other books on general topology, for example, the Engelking's book. Thanks again for the answer:)2012-08-01