Q:

topological properties (invariants)

Can someone explain me why weight and density are topological properties.
Here standard definitions are used. Namely,
$1)$ Weight w(X), the least cardinality of a basis of the topology of the space X
$2)$ Density d(X), the least cardinality of a subset of X whose closure is X.

A:

If $f: X \rightarrow Y$ is a homeomorphism,then
$D$ is dense in $X$ iff $f[D]$ is dense in $Y$ and $|D| = |f[D]|$
$\mathcal{B}$ is a base for $X$ iff $\mathcal{B}' = \{ f[O]: O \in \mathcal{B} \}$ is a base for $Y$ and $|\mathcal{B}| = |\mathcal{B}'|$.
So a minimal dense set or base can be transported trivially to the homeomorphic space, keeping the size.

