Are cardinal functions topological properties?

Question

As we know that cardinality itself is a topological property. But now i am a bit confused about the other cardinal functions like character, extent, weight, cellularity pseudoextent. I want to know if these all are preserved during homeomorphism. Can anybody explain?

Answer 1

Of course all cardinal functions that are usually considered for topological spaces are topological invariants: they are purely defined in set theory terms and notions that depend on the open or closed sets of $X$. If $X \simeq Y$, then anything in $X$ can be "translated" to $Y$ using a given homeomorphism $h: X \to Y$, e.g. if $X$ has character $\kappa$ and $y \in Y$, then there is a unique $x \in X$ with $h(x)=y$ and if $\mathcal{B}$ is a local base at $x$, $\{h[B]\mid B \in \mathcal{B}\}$ is a local base at $y$ etc. so we can choose one of size $\le \kappa$ in $X$ and this translated local base will be of the same size, so also $\le \kappa$ etc. So the character of $Y$ is also $\kappa$ (not smaller, because we can also go back and we'd contradict the minimality of $\kappa$ for $X$). As to cellularity: the images of a disjoint open family in $X$ are also a disjoint open family in $Y$ and vice versa, so maximal sizes of them are also preserved. For all cardinal functions we can hold similar arguments, but in all papers it's usually omitted as being to obvious to mention: if the definition only uses set theory and (properties derived from) open sets, it's obviously a topological invariant.

The chapter by Hodel on cardinal functions in the Handbook of Set-theoretic Topology has a nice first overview of the most common ones and their interrelations (a diagram at the end of the chapter), Engelking's General Topology has a series of exercises on them (if you're self-studying) and the two books by Juhász on Cardinal Functions in Topology are seminal classics on this topic.