# Large discrete subsets of connected $T_2$-spaces

If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete.

Let $\kappa$ be an infinite cardinal. Is there a connected $T_2$-space $(X,\tau)$ and a discrete subset $S\subseteq X$ such that $|X| = |S| = \kappa$?

• Assume that $H$ is a Hilbert space of dimension $\kappa$. That is the cardinality of a maximal orthonormal set is $\kappa$. Let $B$ be this Hilbert space base. Am I mistaken to think that the union of all finite dimensional subspace spaned by elements of $B$ is a possible candidate for your question, if $\kappa$ is not countable? – Ali Taghavi Jan 6 '17 at 14:43
• @AliTaghavi: That works for $\kappa \ge \mathfrak{c}$, of course. And in fact, in that case, you can just take $X=H$, since $H$ itself also has cardinality $\kappa$ (exercise). – Nate Eldredge Sep 9 '18 at 19:25

For any cardinal $\kappa$ at least the size of the continuum, the "really long line" of length $\kappa$ is an example.

This space, let's call it $L_\kappa$, is defined as follows. Begin with the ordinal $\kappa$ with its usual order topology. Then, for any $\alpha \in \kappa$, connect $\alpha$ and $\alpha+1$ with a copy of the unit interval.

$L_\kappa$ obviously has cardinality $\kappa$, and a discrete subspace of size $\kappa$ is given by the set of all successor ordinals.

If $\kappa < \mathfrak{c}$, then you can get an example by modifying the construction of $L_\kappa$: simply replace the unit interval with a countable connected Hausdorff space. (It's not obvious, but countable connected Hausdorff spaces do exist.)

• Another (to my eyes less exotic) example of a countable connected Hausdorff space is the arithmetic progression topology on the integers, due to Golomb jstor.org/stable/2309340?seq=1#page_scan_tab_contents. – Todd Trimble Jan 6 '17 at 15:45
• @ToddTrimble: Thanks for pointing that out. I think Furstenburg's original article used a finer topology that was homeomorphic to the rationals, but this variant of it (which I don't think I've seen before) is very nice. I agree that it's less exotic than the example I link to (which I found just by doing an internet search for "countable connected Hausdorff space".) – Will Brian Jan 6 '17 at 15:52

If you want a closed discrete set take the metric hedgehog with $\kappa$ many spines, where $\kappa$ is the desired cardinality. If $\kappa<\mathfrak{c}$ replace $[0,1]$ by a countable connected Hausdorff space, as in Will Brian's answer.