Space Suggestion

Fix an arbitrary set $B$ of cardinality greater than $2^{\mathfrak c}$ disjoint from $\mathbf N$. Let $X=B\cup \mathbf N$, where open sets are the finite subsets of $\mathbf N$ and the sets containing a cofinite subset of $\mathbf N$. Equivalently, the closed subsets of X are the sets with finite intersection with $\mathbf N$ or containing $B$. (This is analogous to the one-point compactification of the natural numbers, only with many copies of the point at infinity.)

Rationale

Given as Example 2.10 in https://www.sciencedirect.com/science/article/pii/S0166864117300494

There is no example of a large separable space in the database so far.

Relationship to other spaces and properties

This space is separable, T_1, first-countable, locally compact Hausdorff, of cardinality greater than 2^c (in fact, arbitrarily large).