I'd like to know the separation properties (is it Hausdorff, regular or normal) for the following bases.
Suppose $X$ is a topological space
(1) $\mathcal{B}=\{[a,b):a<b; a,b \in R\}$
Hausdorff: if we take a point $a$ and $b,$ with $a < b$
there is a disjoint open set $[a,b)$ and $[b, b+1)$ such that $a \in [a,b) $and $b\in [b, b+1).$
Since their intersection is empty, then it is Hausdorff.
Regular: If $a \in X$ and $A$ is closed subset of $X$ with $a \notin A,$
there exist an open set $[a,b)$ with $[a,b) \subset (X \setminus A)$
then $[a,b)$ and $X\setminus[a,b)$ are disjoint open set with $a \in [a,b)$ and $A \subset X\setminus[a,b)$
therefore $X$ is regular.
Normal: Let a,b $\in R$, with a < b, there is c $\in R$ such that a < c < b
Let A and B be disjoint closed set in X with a $\in$ A and b $\in$ B
a $\in$ A = [a,c) and b $\in$ B = [c, b+1)
[a,c) $\cap$ [c, b+1) is empty.
There exist open sets
U = $\cup$ [a,c) for a $\in$ A
and V = $\cup$ [c,b+1) for b $\in$ B
which are disjoint with A $\subset$ U and B $\subset$ V.
Hence X is normal.
(2) $\mathcal{B} = \{(a,b): a<b; a,b \in \mathbb{R}\}$
(3) $\mathcal{B} = \{(a,\infty): a \in \mathbb{R}\}$
Please Is (1) correct? Anyone can help on (2) and (3)
You're fine for Hausdorffness for (1). Regularity and normality require a bit more work.
(2) is the usual topology, which is metrisable, so all mentioned separattion axioms follow from that.
(3) will fail Hausdorffness, as all non-empty open sets intersect. For similar reasons regularity will fail. Are there even disjoint non-empty closed sets to test normality?