Point Set Topology - Computer Graphics and Geometric Modeling - page 306

compact. Next, assume that n > 1. Again, since A is closed and bounded, A is a closed
subset of a product of intervals X = [a 1 ,b 1 ] ¥ [a 2 ,b 2 ] ¥ ...¥ [a n ,b n ]. Each interval [a i ,b i ]
is compact by the Heine-Borel theorem and so X is compact by the Tychonoff product
theorem. Theorem 5.5.2 in turn implies that A is compact and Theorem 5.5.6 is proved.

The special case of Theorem 5.5.6 where n is 1 is usually referred to as the Heine-
Borel-Lebesgue theorem.
Note that a closed and bounded subset of an arbitrary metric space need not be
compact. For example, by Theorem 5.2.5 we can give R n a bounded metric that
induces the standard topology, but R n is obviously not compact. It follows that Euclid-
ean space R n is special when it comes to Theorem 5.5.6. Although being closed and
bounded is equivalent to being compact for subspaces of R n , the importance of the
latter concept is that it clearly shows that we are dealing with an intrinsic property
of a space that has nothing to do with any particular imbedding in R n .
There is a generalization of Theorem 5.5.6 to metric spaces. It asserts that a subset
of a complete metric space ( X ,d) is compact if and only if it is d-closed and “d-totally
bounded.” See [Eise74]. We also have the following:

5.5.7. Theorem. A metrizable space X is compact if and only if each infinite subset
of X has a limit point in X . A compact metrizable space is complete.

Proof. See [Eise74].

From Theorems 5.5.2 and 5.5.3 we get the classical Bolzano-Weierstrass theorem:
Every bounded infinite set of real numbers has a limit point in R .

5.5.8. Theorem. Let f : X Æ Y be a continuous map between topological spaces. If
X is compact, then so is f( X ).

Proof. Every open cover of f( X ) pulls back to an open cover of X, which has a finite
subcover so that the corresponding open sets in the cover for f( X ) provide a finite sub-
cover of f( X ). See the proof of Theorem 4.2.11.

We shall see that Theorem 5.5.8 has many applications because, for example, lots
of well-known spaces are quotient spaces of compact spaces. Here are three corol-
laries.

5.5.9. Corollary. Compactness is a topological property, that is, if X is homeomor-
phic to Y and if X is compact, then so is Y .

Proof. Clear.

5.5.10. Corollary. Let f : X Æ Y be a continuous map between topological spaces.
Assume that X is compact and Y is Hausdorff. If f is one-to-one and onto, then f is a
homeomorphism.

Proof. It clearly suffices to show that f is a closed map. Let A be a closed subset of
X . Theorem 5.5.2 implies that A is compact and therefore f( A ) is compact by Theorem
5.5.8. Finally, Theorem 5.5.3 implies that f( A ) is closed.