I wish to prove that the intersection of closed sets is closed. However, all proofs that I have come across use Morgan's Theorem, which we have not seen in class (and therefore cannot use).
I was thinking that maybe I could use something like the fact that a closed set is one which contains all of its border points, so intersecting several closed sets should also include all of its border points. However I am not sure if that it is valid in this scenario.
Best Answer
Suppose $C_i, i \in I$ is a family of closed subsets, and let $C$ be their intersection. Suppose $x \in C'$ (a limit point of $C$). Then as $C \subseteq C_i$ for each $i$, $x$ is in $C'_i$ for all $i$, and as each $C_i$ is closed, we know that for each $i$, $C'_i \subseteq C_i$ and thus $x \in \bigcap_i C_i = C$. As $C$ contains all its limit points, $C$ is closed.