What is a weight of space? I came across a paper (sourced below), where the author was talking about a weight of a topological space. In particular, they said the space is "of a countable weight". What does this mean in your opinion? I haven´t found any definition or other reference to weight in the article.
Thank you!



The paper: Topological compactifications
 A: Just to close the question: weight is just one of the many so-called cardinal invariants of a topological space.
Given a space $X$ we can consider the set of all bases for the topology of $X$ and this itself forms a set and so it has a member of minimal cardinality (as cardinal numbers are well-ordered) and this minimal size of a base for $X$ is called the weight of $X$, or $w(X)$. By common convention we round it up to $\aleph_0$ (or countable) in the case where this is a finite number (which happens for finite spaces and some trivial topologies on infinite sets $X$).
So saying the weight of $X$ is countable is just a fancy way of saying that $X$ has at least one countable (or finite) base for its topology, which is also called "$X$ is second countable" or $X$ is $C$-II in some texts.
Another common cardinal invariant is the density of $X$, $d(X)$, which is the size of a minimal dense set, so $d(X)$ countable means that $X$ is separable, etc. See this encyclopaedia article for more examples.
Defining these cardinal numbers allows us to succinctly state facts like $|X|\le 2^{\chi(X)l(X)}$, which is a classical theorem due to Arkhangel'skij. Many papers use these invariants as handy shortcuts.
