# Definition of topological space & open sets

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?

• How about this example where $A\cap B\ne\emptyset$ and $A\cup B\ne X$: $X=\{1,2,3,4\}$, $\mathcal T=\{\emptyset,\{1\},\{2\},\{1,2\},\{1,2,3\},\{1,2,3,4\}\}$, $A=\{1,2\}$, $B=\{2,3\}$ – J. W. Tanner Jun 7 '20 at 11:10

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 $$B \ \notin \ \tau$$.

• I would recommend denoting the empty set as $\emptyset$ rather than $\phi$. – Zest Jun 4 '20 at 16:24
• You can use either \emptyset $\emptyset$ or \varnothing $\varnothing$. – Paul Sinclair Jun 5 '20 at 3:02
• True that! Thanks for the recommendations. – matumath Jun 5 '20 at 14:53

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 $$B\not\in\mathcal T$$.