Exercise 2, § 4

Let $X$ be a topological space which satisfies axiom $(O_V)$.

a) Show that the relation $R : \overline{\{x\}} \cap \overline{\{y\}} \neq \varnothing$ between two points $x, y$ of $X$ is equivalent to the relation $R' : "f(x) = f(y)"$ for every real-valued continuous function $f$ on $X$; hence show that $R$ is an equivalence relation on $X$.

b) Show that the quotient space $Y = X/R$ is normal and that, if $\varphi : X \to Y$ is the canonical mapping, every continuous real-valued function $f$ on $X$ can be written in the form $f = g \circ \varphi$, where $g$ is a continuous real-valued function on $Y$.