Exercise 14, § 2

Let $X$ be a set and let $\mathfrak{c}$ be an infinite cardinal such that $\mathfrak{c}<$ Card(X).

a) Let $\mathscr{U}_{\mathfrak{c}}$ be the set of parts V of X such that Card(X-V) $\leqslant \mathfrak{c}$ or $V =\emptyset$. Exercise prove that $\mathscr{U}_{\mathfrak{c}}$ is a topology on X.

b) Now prove that the set X, endowed with this topology, is connected and locally connected, but that its arcwise connected components are reduced to one element.

c) Now prove that the cone C(X) is arcwise connected, locally connected, but is not locally arcwise connected. This demonstrates that in corollary 2 of III, p. 267, one cannot omit the assumption that the space is locally compact and metrizable.