Example of topological space $(X,\tau)$ is Hausdorff but not discrete.

Question

Would you give me an example of topological space $(X,\tau)$ such that $X$ is enumerable, and $\tau$ is Hausdorff but not discrete. How can I construct this?

Answer 1

$\mathbb{Q}\subseteq\mathbb{R}$ in the subspace topology is Hausdorff, but not discrete.

Answer 2

Let $a_n$ be a sequence of real numbers converging to $a$, then $\{a_n\mid n\in\Bbb N\}\cup\{a\}$ with the subspace topology is not discrete because $\{a\}$ is not an open set.

Answer 3

1 Appert Space

2 Arens Square

3 Arens-Fort Space

4 Closed Countable Ordinal Space

5 Countable Fort Space

6 Evenly Space Integer Topology

7 Gustin’s Sequence Space

8 Irrational Slope Topology

9 Irregular Lattice Topology

10 Minimal Hausdorff Topology

11 Open Countable Ordinal Space

12 Prime Integer Topology

13 Relatively Prime Integer Topology

14 Roy’s Lattice Space

15 Roy’s Lattice Subspace

16 Single Ultrafilter Topology

17 The p-adic Topology on the Integers

18 The Rational Numbers

More details on examples see here.