Exercise 30, § 9

One says that a topological space X is locally quasi-compact if every point of X admits a fundamental system of quasi-compact neighbourhoods.

a) Now prove that in a locally quasi-compact space, every quasi-compact set admits a fundamental system of quasi-compact neighbourhoods.

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

*c) On the Euclidean closed disk B: $\|x\| \leq 1$ in the plane $\mathbf{R}^2$, consider the topology $\mathfrak{T}$ finer than the topology $\mathfrak{T}_0$ induced by that of $\mathbf{R}^2$, such that for $\mathfrak{T}$ the neighbourhood filter of a point $x \neq (1, 0)$ of B is identical with the neighbourhood filter of this point for $\mathfrak{T}_0$, but a fundamental system of neighbourhoods of $(1, 0)$ is formed by the unions of $\{(1, 0)\}$ and the trace on the open disk $B_0: \|x\| < 1$ of an open disk with center $(1, 0)$. Let Y be the set B endowed with the topology $\mathfrak{T}$; consider in Y the equivalence relation whose classes are the set S of the $x \neq (1, 0)$ such that $\|x\| = 1$, and the subsets of Y reduced to a point not in S. Now prove that the quotient space $X = Y/R$ is quasi-compact but not locally quasi-compact.