[alg.top]
Cell complexes are always path-connected.

[alg.top]
Standard (cubic lattice) CW complex structure on $\mathbb{R}^n$: Lattice of 0-cells, join all by 1-cells, then add 2-cells, 3-cells etc.

Check this rigorously.

[alg.top]
$X$ contractible $\ \nRightarrow$ $X$ deformation retracts to a point

But the converse is true?