Introduction To Topology Mendelson Solutions //top\\ Instant

If you are interested in learning more about topology, here are some further resources:

A common early exercise in Mendelson (Chapter 2) involves proving a set in a metric space is open using the "Open Ball" definition. Topology textbook with a solution manual Introduction To Topology Mendelson Solutions

: A collaborative project providing TeX-formatted solutions to exercises, specifically organized by chapter and problem number. If you are interested in learning more about

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. and hence $f(X)$ is compact.