finite and countable discrete spaces

Theorem 1.

Suppose $X\mathrm{\ne }\mathrm{\varnothing }$ is equipped with the discrete topology.

1. If $X$ if finite, then $X$ is homeomorphic to $\{1,\mathrm{\dots },n\}$for some $n\ge 1$.

2. If $X$ if countable, then $X$ is homeomorphic to $\mathbb{Z}$.

Here, $\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}n\mathrm{\}}$ and $\mathrm{Z}$ are endowed with the discrete topology(or, equivalently, the subspace topology from $\mathrm{R}$).

Proof.

The first claim will be proven. If

$$X=\{a_1,\mathrm{\dots },a_n\}$$

let $\mathrm{\Phi }:\{1,\mathrm{\dots },n\}\to X$ be

$$\mathrm{\Phi }(i)=a_i,i=1,\mathrm{\dots },n.$$

Since $\mathrm{\Phi }$ is a bijection, it is a homeomorphism.

The proof of the second claim is to that of the first.∎