A Mendelson solutions guide worth its salt will include this classic counterexample with a detailed explanation of why ( xy=1 ) is closed (pre-image of ( 1 ) under continuous multiplication) and why the punctured line is not closed.
Let $X$ be a compact topological space and let $f: X \to Y$ be a continuous function. Let $U_\alpha$ be an open cover of $f(X)$. Then, $f^-1(U_\alpha)$ is an open cover of $X$. Since $X$ is compact, there exists a finite subcover $f^-1(U_\alpha_i)$. This implies that $U_\alpha_i$ is a finite subcover of $f(X)$, and hence $f(X)$ is compact. Introduction To Topology Mendelson Solutions
: Mendelson uses metric spaces in Chapter 2 as a bridge. By introducing limits, continuity, and open sets in the context of distance, he provides a "crutch" for students before removing it to introduce general topological spaces in Chapter 3. A Mendelson solutions guide worth its salt will