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**Showcase_22** $\displaystyle (X,T')$ compact gives:

Let $\displaystyle U$ be any open cover of $\displaystyle (X,T')$. Then there exists a finite subcover of $\displaystyle U$, say $\displaystyle U_1=\{u_1, \ldots, u_n\}$ such that $\displaystyle X \subset U_1 \subset \bigcup_{i=1}^n u_i$.

Each $\displaystyle u_i$ is open (is the "finite subcover" always open?) so each $\displaystyle u_i \in T'$.

From here I get stuck. If we consider cases, if every $\displaystyle u_i \in T \subset T'$ then i'm done! However, if $\displaystyle u_i \in T'-T$ then i'm not really sure what to do. I don't even think this is possible since $\displaystyle u_i$'s are open so should be in $\displaystyle T$ anyway.

I'd really appreciate some help!