Introduction To Topology Mendelson Solutions <Top 50 RECOMMENDED>

Foundations, Logic, and Countability.

Show that ( f: \mathbbR \to \mathbbR ), ( f(x)=x^2 ) is continuous (usual topology) using ε-δ. Introduction To Topology Mendelson Solutions

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. Foundations, Logic, and Countability