Exercises 3.4 Problem set 5, due 4 December 2021
Let \((X, \tau) \) be a topological space. Define \(X^{\ast} = X \cup \{\infty\} \text{,}\) where \(\infty \notin X \text{.}\) We now define a topology \(\tau^{\ast} \) as follows. Let \(S \subseteq X^{\ast} \) be a subset. For any point \(x \in X \text{,}\) we declare that \(x \) is close to \(S \) if and only if \(x \) is close to \(S \cap X \text{.}\) We also declare that \(\infty \) is close to \(S \) if and only if, for every closed and compact subspace \(Z \subseteq X \text{,}\) the set \(S \) intersects the complement \(X^{\ast} \setminus Z\text{.}\)
1.
Characterize the open and closed subsets of \(X^{\ast} \text{.}\)
2.
Prove that \(X^{\ast}\) is compact. We call \(X^{\ast}\) the one-point compactification of \(X\text{.}\)
3.
Assume that \(X\) is hausdorff and locally compact: that is, every point is contained in an open neighborhood whose closure is compact. Prove that \(X^{\ast}\) is hausdorff.
Let \(X \) and \(Y\) be topological spaces. A map \(f \colon X \to Y \) is proper if and only if, for every compact subspace \(K \subseteq Y \text{,}\) the preimage \(f^{-1}K \subseteq X \) is compact as well.
4.
Assume that \(X \) and \(Y \) are locally compact hausdorff, and that \(f \colon X \to Y \) is continuous. Show that the following are equivalent.
The map \(f\) is proper.
For any topological space \(Z\text{,}\) the map \(\id\times f \colon Z \times X \to Z \times Y \) is closed.
The map \(f\) is closed, and for every point \(y\in Y\text{,}\) the preimage \(f^{-1}\{y\}\) is compact.
5.
Which of the following maps is proper? (For definiteness, please assume that all the topological spaces below are locally compact and hausdorff.) You do not need to prove that your computation is correct.
a continuous map \(X \to Y \) in which \(X \) is compact and \(Y \) is hausdorff;
the inclusion map of an open subspace \(U \subseteq X\text{;}\)
the inclusion map of a closed subspace \(Z \subseteq X \text{;}\)
the map \(\RR \to \RR\) given by \(t \mapsto \exp(t)\text{;}\)
the product of any family of proper continuous maps;
for \(n\geq 1\text{,}\) the quotient map \(\CC^{n+1}\setminus \{0\} \to \PP^n_{\CC}\text{;}\)
for \(n\geq 1\text{,}\) the quotient map \(S^{2n+1} \to \PP^n_{\CC}\text{;}\)
6.
Keeping our notations as above, show that if \(f \) is proper and continuous, then it extends to a continuous map \(f^{\ast} \colon X^{\ast} \to Y^{\ast} \) that carries \(\infty\) to \(\infty\text{.}\)
7.
Give an example of two locally compact hausdorff spaces \(X \) and \(Y \) and a continuous map \(f \colon X \to Y \) that does not extend to a continuous map \(f^{\ast} \colon X^{\ast} \to Y^{\ast}\text{.}\)