Empty Topological Space is Locally Euclidean Space of any Dimension

Theorem

Let $T = \struct {\O, \set \O}$ be the empty topological space.

Then $T$ is locally euclidean space for every dimension.

Proof

Recall the definition of Locally Euclidean Space:

Let $M$ be a topological space.

Let $d \in \N$ be a natural number.

Then $M$ is a locally Euclidean space of dimension $d$ if and only if each point in $M$ has an open neighbourhood which is homeomorphic to an open subset of Euclidean space $\R^d$.

This is vacuously true for the empty set for every $d \in \N$.

$\blacksquare$