Need example for a topological space that isn't connected, but is compact

Question

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?

Answer 1

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

Answer 2

$\pi$-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

Answer 3

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 $Y \times \{1\}$.