CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
For $n \in \mathbb{N}$ a natural number, write $\mathbb{R}^n$ for the Cartesian space of dimension $n$. The Euclidean topology is the topology on $\mathbb{R}^n$ characterized by the following equivalent statements
it is the metric topology induced from the canonical structure of a metric space on $\mathbb{R}^n$ with distance function given by $d(x,y) = \sqrt{\sum_{i = 1}^n (x_i-y_i)^2}$;
an open subset is precisely a subset such that contains an open ball around each of its points;
it is the product topology induced from the standard topology on the real line.
Two Cartesian spaces $\mathbb{R}^k$ and $\mathbb{R}^l$ (with the Euclidean topology) are homeomorphic precisely if $k = l$.
A proof of this statement was an early success of algebraic topology.