Definition of topological space & open sets

Question

I am just getting into topology, and I have a doubt regarding open sets.

Let $(X, \mathcal{T})$ be a topological space. Given an open set of $X$, $A$, and subset of $X$, $B$ such that

$$A\cap B \in \mathcal{T}$$ $$A\cup B \in \mathcal{T}$$

Can I conclude that $B$ is also an open set? That is, if I have an arbitrary set of $X$ whose intersection and union with an open set are themselves open sets, does this imply the arbitrary set is also open?

Answer 1

No. It does not imply that the set is open. For instance:

Let $X= \left\lbrace a, b, c\right\rbrace$ and consider the topological space $(X, \tau)$ where $\tau=\left\lbrace\varnothing, X, \left\lbrace a \right\rbrace\right\rbrace$.

Let $A=\left\lbrace a \right\rbrace$, let $B=\left\lbrace b, c \right\rbrace$.

Then,

$A \cup B= X \ \in \ \tau $,

$A \cap B= \varnothing \ \in \ \tau$.

However $B \ \notin \ \tau$.

Answer 2

No. Consider the usual topology on $X=\mathbb R$ with $A=(-\infty,0)$ and $B=[0,\infty)$.

$A\cap B=\emptyset\in\mathcal T$, $ A\cup B=X\in\mathcal T$, $A\in\mathcal T$, and $B\not\in\mathcal T$.