# finite and countable discrete spaces

###### Theorem 1.

Suppose $X\neq\emptyset$ is equipped with the discrete topology.

1. 1.

If $X$ if finite, then $X$ is homeomorphic to $\{1,\ldots,n\}$ for some $n\geq 1$.

2. 2.

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

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

###### Proof.

The first claim will be proven. If

 $X=\{a_{1},\ldots,a_{n}\}$

let $\Phi\colon\{1,\ldots,n\}\to X$ be

 $\Phi(i)=a_{i},\quad i=1,\ldots,n.$

Since $\Phi$ 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 FiniteAndCountableDiscreteSpaces 2013-03-22 15:17:11 2013-03-22 15:17:11 matte (1858) matte (1858) 9 matte (1858) Theorem msc 54-00