2018-57.md

course	course_year	question_number	tags	title	year
Metric and Topological Spaces	IB	57	IB 2018 Metric and Topological Spaces	Paper 4, Section II, E	2018

Let $X={2,3,4,5,6,7,8, \ldots}$ and for each $n \in X$ let

$$U_{n}={d \in X \mid d \text { divides } n} .$$

Prove that the set of unions of the sets $U_{n}$ forms a topology on $X$.

Prove or disprove each of the following:

(i) $X$ is Hausdorff;

(ii) $X$ is compact.

If $Y$ and $Z$ are topological spaces, $Y$ is the union of closed subspaces $A$ and $B$, and $f: Y \rightarrow Z$ is a function such that both $\left.f\right|{A}: A \rightarrow Z$ and $\left.f\right|{B}: B \rightarrow Z$ are continuous, show that $f$ is continuous. Hence show that $X$ is path-connected.