Exercise 10, § 1

Let $X$ be a completely regular space, let $K$ be a compact subset of $X$, and let $V$ be a neighbourhood of $K$ in $X$.

a) Show that there is a continuous mapping of $X$ into $[0, 1]$ which is equal to 1 on $K$ and equal to 0 on $\mathbf{C}V$ [use (OIV), covering $K$ by a finite number of suitably chosen neighbourhoods].

b) Let $X'$ be the quotient space of $X$ obtained by identifying all the points of $K$. Show that $X'$ is completely regular.