How to construct counterexamples of spaces which satisfy some separation axioms?

Question

I want to know some counterexamples of spaces which satisfy some separation axioms.For example, the counterexample for a space which doesn't satisfy $T_0$,$T_3$ and $T_4$ axioms.

Is there a more efficient way to construct these counterexamples?

Thank you!

Answer 1

Consider the following statement

If topological spaces $(X_i,\mathcal{T}_i)$ satisfy $\mathrm{T}_k$ axiom for all $i\in I$, where $k\in\{0,1,2,3,4\}$, then $\sqcup_{i\in I}(X_i,\mathcal{T}_i)$ with the disjoint union topology also satisfies $\mathrm{T}_k$ axiom.

Then you will have something like this. The examples.

Thus you only need to consider the cofinite topological spaces, the Sierpinski topological space, the trivial topological space, the Moore topological space, the metric space, the half-disc topological space.

The above topological spaces could be easily found in any textbook, or on wikipedia.