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. ∎
Title  finite and countable discrete spaces 

Canonical name  FiniteAndCountableDiscreteSpaces 
Date of creation  20130322 15:17:11 
Last modified on  20130322 15:17:11 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  9 
Author  matte (1858) 
Entry type  Theorem 
Classification  msc 5400 