Intersection of Topologies is Topology

Theorem
Let $\left({\tau_i}\right)_{i \in I}$ be a family of topologies for a set $X$.

Then $\displaystyle \bigcap_{i \in I} {\tau_i}$ is also a topology for $X$.

$X \in \tau$
By the definition of a topology:
 * $\forall i \in I: X \in \tau_i$

Thus by definition of set intersection we have that:
 * $\displaystyle X \in \bigcap_{i \in I} {\tau_i}$

Union in $\tau$
Let $\left({U_j}\right)_{j \in J}$ be a family of open sets of $\displaystyle \bigcap_{i \in I} {\tau_i}$.

Thus for all $i \in I$, we have by definition of set intersection that:
 * $\forall j \in J: U_j \in \tau_i$

Since $\tau_i$ is a topology for every $i \in I$, by definition we have:
 * $\displaystyle \forall i \in I: \bigcup_{j \in J} {U_j} \in \tau_i$

Therefore we have:
 * $\displaystyle \bigcup_{j \in J} {U_j} \in \bigcap_{i \in I} {\tau_i}$

Intersection in $\tau$
Let $U_1, U_2$ be open sets of $\displaystyle \bigcap_{i \in I} {\tau_i}$.

Then by definition of set intersection:
 * $\forall i \in I: U_1, U_2 \in \tau_i$

Since $\tau_i$ is a topology for each $i \in I$, we obtain that:
 * $\forall i \in I: U_1 \cap U_2 \in \tau_i$

Therefore we have:
 * $\displaystyle U_1 \cap U_2 \in \bigcap_{i \in I} {\tau_i}$

Thus, by definition, $\displaystyle \bigcap_{i \in I} {\tau_i}$ is a topology.