General Intersection Property of Topological Space

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Theorem

Let $\left({S, \tau}\right)$ be a topological space.

Let $S_1, S_2, \ldots, S_n$ be open sets of $\left({S, \tau}\right)$.


Then:

$\displaystyle \bigcap_{i \mathop = 1}^n S_i$

is also an open set of $\left({S, \tau}\right)$.

That is, the intersection of any finite number of open sets of a topology is also in $\tau$.


Conversely, if the intersection of any finite number of open sets of a topology is also in $\tau$, then:

$(1): \quad$ The intersection of any two elements of $\tau$ is an element of $\tau$
$(2): \quad S$ is itself an element of $\tau$.


Proof

Proof by induction:

For all $n \in \N_{>0}$, let $P \left({n}\right)$ be the proposition:

For any sets $S_1, S_2, \ldots, S_n \in \tau$, it follows that $\displaystyle \bigcap_{i \mathop = 1}^n S_i \in \tau$.


Let $\mathbb S$ be any finite subset of $\tau$.


From Intersection of Empty Set, we have that:

$\mathbb S = \varnothing \implies \bigcap \mathbb S = S$

From the definition of a topology, we have that $S \in \tau$.

Hence $P(0)$ is true.


From Intersection of Singleton, we have that:

$\mathbb S = S_1 \implies \bigcap \mathbb S = S_1$

Thus $P(1)$ is trivially true.


Basis for the Induction

$P(2)$ is the case $S_1 \cap S_2 \in \varnothing$, which is our axiom:

The intersection of any two elements of $\tau$ is an element of $\tau$.

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.


So this is our induction hypothesis:

For any sets $S_1, S_2, \ldots, S_k \in \tau$, it follows that $\displaystyle \bigcap_{i \mathop = 1}^k S_i \in \tau$.


Then we need to show:

For any sets $S_1, S_2, \ldots, S_{k+1} \in \tau$, it follows that $\displaystyle \bigcap_{i \mathop = 1}^{k+1} S_i \in \tau$.


Induction Step

This is our induction step:

Consider the set $\displaystyle \bigcap_{i \mathop = 1}^{k+1} S_i$.

This is $\displaystyle \left({\bigcap_{i \mathop = 1}^k S_i}\right) \cap S_{k+1}$.

But from the induction hypothesis, we know that:

$\displaystyle \left({\bigcap_{i \mathop = 1}^k S_i}\right) \in \tau$

So from the base case, it follows that:

$\displaystyle \bigcap_{i \mathop = 1}^{k+1} S_i \in \tau$


So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.


Therefore, for any sets $S_1, S_2, \ldots, S_n \in \tau$, it follows that $\displaystyle \bigcap_{i \mathop = 1}^n S_i \in \tau$.

$\Box$


The converse follows directly from the above.

In particular, the fact that $S$ is itself an element of $\tau$ follows from the definition of Intersection of Empty Set.

$\blacksquare$


Sources

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