# Let X be a topological space. a) If X is Hausdorff and it has an countable base, so X is normal?

Let X be a topological space.

a) If X is Hausdorff and it has an countable base, so X is normal?

b) If X is regular, and if each point of X has a fundamental countable neighborhood system and is separable, then is X normal?

Being normal means that X is $$T_1$$ and $$T_4$$.

We say that:

X is $$T_4$$ if for any A, B $$\subset$$ X closed there are U, V $$\subset$$ X open disjoint such that A $$\subset$$ U and B $$\subset$$ V.

X is regular if X is $$T_3$$ and $$T_1$$.

X is $$T_3$$ iff for each x $$\in$$ X and each F $$\subset$$ X closed such that x $$\notin$$ F there are U, V $$\subset$$ X open disjoint such that x $$\in$$ U and F $$\subset$$ V.

If X is $$T_3$$ is $$T_2$$ and $$T_1$$.

If X is $$T_2$$ is also $$T_1$$.

if X is Separable it contains a subset A where for any point x in X, any neighborhood of x contains at least one point from A and A is countable. (A space is called separable if it contains a countable, dense subset).

a) and b) are false, right?

In both cases, there are $$T_2$$ spaces that we want to prove are normal.

Being $$T_2$$ means that U, V $$\subset$$ X open disjoint such that A $$\subset$$ U and B $$\subset$$ V.But I don't see how to find a closed for any A, B, even with the countable base (first case), as in the case of having a fundamental system of countable neighborhood and being separable (second case).

• A book with hundreds of counterexamples in topology is called (guess what) Counterexamples in Topology by Steen & Seebach. An On-line resource is $\pi$-base at topology.jdabbs.com Nov 16, 2020 at 22:03

The first has the $$\Bbb R_K$$ as a counterexample.