Exercise 2.2.6

Question

Let $X$ be a topological space, let $S$ and $T$ be subsets of $X$, and let $A$ be a subset of $S \cap T$ that is relatively open in $S$ and in $T$. Is $A$ relatively open in $S \cup T ?$ Justify your answer.

Elu Thingol · Accepted Answer

\begin{proof}
Unwrapping the definitions, we obtain an open set $U$ in $X$ such that $A = U \cap S$ and an open set $V$ in $X$ such that $A = V \cap T$. We then have, by applying the distributive law for set union and intersection several times,
\begin{align*}
(S \cup T) \cap (U \cap V) &= (S \cap U \cap V) \cup (T \cap U \cap V)\
&= (A \cap V) \cup (A  \cap U)\
&= A \cap (U \cup V) = A.
\end{align*}
Since $U \cap V$ is open in $X$, $A$ must be open in $S \cup T$.
\end{proof}

Exercise 2.2.6

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Exercise 2.2.6

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