?

Topology

Free Version

Upgrade subject to access all content

Moderate

Examples of Compact Spaces

TOPO-DLJTJZ

Which of the following topological spaces $(X,\tau)$ is compact?

A

$X=\bigsqcup_{k\in\mathbb{Z},k > 0} [0,1/k]/\sim$, where $\sim$ identifies all the end points of all intervals $[0,1/k]$. $\tau$ is the quotient topology.

B

$X=[a,b]$, where $a < b$ are two points on the long line.

C

$X$ is an uncountable set and $\tau$ consists of the empty set as well as sets with countable complement.

D

$X=\mathbb{R}$ with discrete topology.