# Equivalence of Definitions of Closed Set

## Theorem

The following definitions of the concept of **Closed Set** in the context of **topology** are equivalent:

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

### Definition 1

**$H$ is closed (in $T$)** if and only if its complement $S \setminus H$ is **open in $T$**.

That is, $H$ is **closed** if and only if $\paren {S \setminus H} \in \tau$.

That is, if and only if $S \setminus H$ is an element of the topology of $T$.

### Definition 2

**$H$ is closed (in $T$)** if and only if every limit point of $H$ is also a point of $H$.

That is, by the definition of the derived set:

**$H$ is closed (in $T$)**if and only if $H' \subseteq H$

where $H'$ denotes the derived set of $H$.

## Proof

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

### Definition 1 implies Definition 2

Let $H$ be a closed set of $T$ by definition 1.

Let $H^\complement$ denote the relative complement of $H$ in $S$.

By definition of closed set in $T$:

- $H^\complement$ is open in $T$

From Set is Open iff Neighborhood of all its Points:

- $\forall x \in S: x \notin H \implies H^\complement$ is a neighborhood of $x$.

By definition of limit point:

- $\forall x \in S: x \notin H \implies x$ is not a limit point of $H$

Thus $H$ is a closed set of $T$ by definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $H$ be a closed set of $T$ by definition 2.

Then by definition: $\forall x \in S: x \notin H \implies x$ is not a limit point of $H$

By definition of limit point: $\forall x \in S: x \notin H \implies H^\complement$ is a neighborhood of $x$

By Set is Open iff Neighborhood of all its Points: $H^\complement$ is open in $T$

Thus $H$ is a closed set of $T$ by definition 1.

$\blacksquare$

## Sources

- 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: Limit Points

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- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (previous) ... (next): $2.23$