# Topology Defined by Closed Sets

## Theorem

Let $S$ be a set.

Let $\tau$ be a set of subsets of $S$.

Then $\tau$ is a topology on $S$ if and only if:

- $(1): \quad$ Any intersection of arbitrarily many closed sets of $S$ under $\tau$ is a closed set of $S$ under $\tau$
- $(2): \quad$ The union of any finite number of closed sets of $S$ under $\tau$ is a closed set of $S$ under $\tau$
- $(3): \quad S$ and $\O$ are both closed sets of $S$ under $\tau$

where a closed set $V$ of $S$ under $\tau$ is defined as a subset of $S$ such that $S \setminus V \in \tau$.

## Proof

From the definition, if $V$ is a closed set of $S$, then $S \setminus V$ is an open set of $S$.

Let $\mathbb V$ be any arbitrary set of closed sets of $S$.

Then by De Morgan's Laws: Difference with Intersection, we have:

- $\ds S \setminus \bigcap \mathbb V = \bigcup_{V \mathop \in \mathbb V} \paren {S \setminus V}$

First, let $\tau$ be a topology on $S$.

We have that:

- Intersection of Closed Sets is Closed in Topological Space
- Finite Union of Closed Sets is Closed in Topological Space
- By Open and Closed Sets in Topological Space, $\O$ and $S$ are both closed in $S$.

Thus, the properties as listed above hold.

$\Box$

Suppose the properties:

- $(1): \quad$ Any intersection of arbitrarily many closed sets of $S$ under $\tau$ is a closed set of $S$ under $\tau$
- $(2): \quad$ The union of any finite number of closed sets of $S$ under $\tau$ is a closed set of $S$ under $\tau$
- $(3): \quad S$ and $\O$ are both closed sets of $S$ under $\tau$.

all hold.

That means $\ds \bigcap \mathbb V$ is closed.

So $\ds S \setminus \bigcap \mathbb V = \bigcup_{V \mathop \in \mathbb V} \paren {S \setminus V}$ is open.

Thus we have that the union of arbitrarily many open sets of $S$ under $\tau$ is an open set of $S$ under $\tau$.

Similarly, we deduce that the intersection of any finite number of open sets of $S$ under $\tau$ is an open set of $S$ under $\tau$.

By Open and Closed Sets in Topological Space, $\O$ and $S$ are both open in $S$.

So $\tau$ is a topology on $S$.

$\blacksquare$

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets: Theorem $2$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Proposition $3.7.4$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction