Loading web-font TeX/Math/Italic

The Stacks project

Combination of [I, p. 75, Lemme 1, Bourbaki] and [I, p. 76, Corrolaire 1, Bourbaki].

Lemma 5.17.3. A topological space X is quasi-compact if and only if the projection map Z \times X \to Z is closed for any topological space Z.

Proof. (See also remark below.) If X is not quasi-compact, there exists an open covering X = \bigcup _{i \in I} U_ i such that no finite number of U_ i cover X. Let Z be the subset of the power set \mathcal{P}(I) of I consisting of I and all nonempty finite subsets of I. Define a topology on Z with as a basis for the topology the following sets:

  1. All subsets of Z\setminus \{ I\} .

  2. For every finite subset K of I the set U_ K := \{ J\subset I \mid J \in Z, \ K\subset J \} ).

It is left to the reader to verify this is the basis for a topology. Consider the subset of Z \times X defined by the formula

M = \{ (J, x) \mid J \in Z, \ x \in \bigcap \nolimits _{i \in J} U_ i^ c)\}

If (J, x) \not\in M, then x \in U_ i for some i \in J. Hence U_{\{ i\} } \times U_ i \subset Z \times X is an open subset containing (J, x) and not intersecting M. Hence M is closed. The projection of M to Z is Z-\{ I\} which is not closed. Hence Z \times X \to Z is not closed.

Assume X is quasi-compact. Let Z be a topological space. Let M \subset Z \times X be closed. Let z \in Z be a point which is not in \text{pr}_1(M). By the Tube Lemma 5.17.1 there exists an open U \subset Z such that U \times X is contained in the complement of M. Hence \text{pr}_1(M) is closed. \square


Comments (0)

There are also:

  • 2 comment(s) on Section 5.17: Characterizing proper maps

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.