Need example for a topological space that isn't connected, but is compact We know the topological space $(R,τ_1)$ is a connected space but it is not compact, $(R,τ_+)$ (which generated by $[a,b[$) is not connected space and it is not compact space, and $(R,τ_{cf})$ is connected and compact space.
Can someone give me an example of a compact space that isn't connected? 
 A: Any finite subset of $\mathbb{R}$ with $2$ or more elements will be compact and not connected because:


*

*Any finite subset of $\mathbb{R}$ will be compact (by the Heine-Borel Theorem a subset of $\mathbb{R}$ is compact if and only if it is closed and bounded, which is true for any finite set)

*Any finite subset of $\mathbb{R}$ with $2$ or more elements won't be connected

A: $\pi$-Base is an online encyclopedia of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following twelve compact spaces that are not connected. You can learn more about any of them by visiting the search result.
Closed Ordinal Space $[0,\Gamma]$ ($\Gamma < \Omega$)
Closed Ordinal Space $[0,\Omega]$
Concentric Circles
Countable Fort Space
Either-Or Topology
Finite Discrete Topology
Maximal Compact Topology
Stone-Cech Compactification of the Integers
The Cantor Set
Tychonoff Plank
Uncountable Fort Space
Uncountable Modified Fort Space
A: Take any two compact, connected topological spaces $X,Y$. Define their discrete union to be the disjoint union $X \coprod Y = (X \times \{0\}) \cup (Y \times \{1\})$ with the topology generated by all sets of the form $U \times \{0\}$ or $V \times \{1\}$ where $U \subset X$ is any open set in $X$ and $V \subset Y$ is any open subset of $Y$. Then $X \coprod Y$ is compact and has two components, one being $X \times \{0\}$ and the other $Y \times \{1\}$.
