Classifying Hausdorff spaces

Question

Can a topological space be Hausdorff and separable, but neither Lindelof nor first countable?

Can a topological space be Hausdorff and Lindelof, but neither separable nor first countable?

Can a topological space be Hausdorff, but not separable, Lindelof, or first countable?

Are all second countable spaces Hausdorff?

Sorry for posting several questions in one post, but I am just trying to figure out how the Hausdorff property relates to Lindelof, Separability, and First Countability in topological spaces.

Answer 1

For questions like this, you probably want to get a copy of the book Counterexamples in Topology, by Steen and Seebach. The "general reference chart" in the back of that book allows you to search the counterexamples by combinations of properties they do and don't satisfy.

Fortunately, this chart (and more) now exists in an online searchable form: $\pi$-Base.

Can a topological space be Hausdorff and separable, but neither Lindelof nor first countable?

Yes: The strong ultrafilter topology.

Can a topological space be Hausdorff and Lindelof, but neither separable nor first countable?

Yes: Many examples, including the ordinal space $[0,\omega_1]$ and the countable complement extension topology (the topology generated by the standard topology on $\mathbb{R}$ together with the countable complement topology on $\mathbb{R}$).

Can a topological space be Hausdorff, but not separable, Lindelof, or first countable?

Yes: Many examples, including the product topology on a product of uncountably many copies of an infinite discrete set.

Are all second countable spaces Hausdorff?

No. Many examples, including the cofinite topology on $\mathbb{N}$.

Answer 2

Looking in large products (easy way to get non-first countable spaces with varying properties):

• $\mathbb{R}^\mathbb{R}$ in the product topology is separable, Hausdorff (even Tychonoff), but not Lindelöf, nor first countable.

• $[0,1]^I$ (product topology) is not separable if $|I| > |\mathbb{R}|$, Hausdorff, compact (so certainly Lindelöf) and not first countable for uncountable $I$.

• $\mathbb{R}^I$ with $|I| > |\mathbb{R}|$ is Tychonoff, not separable, not normal (so certainly not Lindelöf as Lindelöf plus regular implies normal) and not first countable.

The final question on non-Hausdorff spaces is different (most "natural" examples as above are Tychonoff, so Hausdorff), and there even the indiscrete topology suffices as an example, which is trivially second countable.