Finite and finite-dimensional CW complex

Question

A finite CW complex is one with a finite number of cells while a finite dimensional CW-complex is one with no cells of dimension greater than a nonnegative integer $n$, we say in this case that $X$ is an $n$-dimensional CW commplex. Obviously every finite cw complex is finite dimensional, but is there a finite dimensional CW-complex that is not a finite CW complex?

Answer 1

A finite dimensional CW-complex can be an infinite CW-complex ( one with infinitely many cells). For example the space $\mathbb{N}$ has infinitely many $0$-cells, namely the singletons $\{n\}$ for each $n\in \mathbb N$ so $\mathbb N$ is a $0$-dimensional and infinite CW-complex.

A connected example is given by the real numbers $\mathbb R$ which is has infinitely many $1$-cells, namely the intervals $[n,n+1]$ for each integer $n\in \mathbb Z$. So $\mathbb R$ is a $1$-dimensional and infinite CW-complex.

Notice that a CW-complex is finite if and only if it is compact, so being finite is a topological property.