Properties of $A$-inclusion topology.

Question

Let $X$ be a non-empty set and $A\subset X$.Define $\tau$ on $X$ as follows: $U$ is open iff $A\subset U $ or $U=\phi$.We can easily verify that this is a topology on $X$.Now what are the characteristics of this space in terms of separability,Lindelofness,First countablity and second countabilty?

Answer 1

Any non-empty open set contains $A$, so if $a \in A$ (we must assume $A \neq \emptyset$ or $\tau$ degenerates to the discrete topology, and things can be different), then $\{a\}$ is (finite, so countable) and dense. $X$ is thus separable. Note that $X$ is such that $A$ is an indiscrete subspace and $X \setminus A$ is relatively discrete and any subset of $A$ is dense.

Every $x \in X$ has a minimal open neighbourhood $O_x:=\{x\} \cup A$, so $X$ is first countable at all points.

Any base for $(X,\tau)$ must thus contain all sets of the form $\{O_x\mid x \in X\}$, and $O_x = A$ for all $x \in A$, and otherwise these are all distinct. So $(X,\tau)$ is second countable iff $X\setminus A$ is at most countable.

Similarly, by considering the open cover $\{O_x\mid x \in X\}$ of $X$ we can show that likewise, $X$ is Lindelöf iff $X\setminus A_x$ is at most countable, so the same condition as for second countability.

Note that if $X$ will almost never be $T_1$ or $T_2$.