Exercise 29, § 9

A topological space $X$ is said to be locally quasi-compact if each point of $X$ has a fundamental system of quasi-compact neighbourhoods.

a) Show that in a locally quasi-compact space, every quasi-compact subset has a fundamental system of quasi-compact neighbourhoods.

b) Show that every locally closed subspace of a locally quasi-compact space is locally quasi-compact.

*c) On the closed Euclidean disc $B : ||x|| \leq 1$ in the real plane $\mathbf{R}^2$, let $\mathfrak{T}_0$ be the topology induced by that of $\mathbf{R}^2$, and let $\mathfrak{T}$ be the (finer) topology defined as follows: the neighbourhood filter of a point $x \neq (1,0)$ of $B$ is the same for $\mathfrak{T}$ as for $\mathfrak{T}_0$, but for the point $(1,0)$ a fundamental system of neighbourhoods in the topology $\mathfrak{T}$ is formed by the sets

where $D$ is any open disc centred at $(1,0)$ and $B_0$ is the open disc $||x|| < 1$. Let $Y$ be the set $B$ with the topology $\mathfrak{T}$, and let $R$ be the equivalence relation on $Y$ whose equivalence classes are the set $S$ of all $x \neq (1,0)$ in B such that $||x|| = 1$ and the subsets of Y which consist of a single point of B — S. Show that the quotient space $X = Y/R$ is quasi-compact but not locally quasi-compact. \*