Given a set $A$ is there a known way to find a topological space $X$ such that $|A|=|X|<w(X)$?


Here $w(X)$ is the weight of the topological space.


This is clearly impossible for finite sets $A$. We know it is possible for $A=\mathbb Z$ because of the answers to [this question](https://mathoverflow.net/questions/105117/countable-topological-spaces-of-uncountable-weight).


Is there a general construction for any infinite set? Is it even known if such a space exists for every infinite $A$? Has a contruction been found for $\mathbb R$ for example?


Thank you kindly.



Is there a known construction for heavy topologies of all sizes?
Let $U$ be a (free) ultrafilter on an infinite set $X$ which contains only sets of cardinality $|X|$ (that is, it is a normal ultrafilter). Put $\tau = U \cup \{\varnothing\}$; this is clearly a topology. 


There is no base $B$ for $\tau$ with $|B|=|X|$ as every such base must have cardinality at least $|X|^+$ by a diagonal argument.


