Closed set

Definition

A closed set in a topological space

$$(X,\mathcal{J})$$ is a set $$A$$ where $$X-A$$ is open.

Metric space

• Note: as every metric space is also a topological space it is still true that a set is closed if its complement is open.

A subset $A$ of the metric space $(X,d)$ is closed if it contains all of its limit points

For convenience only: recall $x$ is a limit point if every neighbourhood of $x$ contains points of $A$ other than $x$ itself.

Example

$(0,1)$ is not closed, as take the point $0$.

Proof

Let $N$ be any neighbourhood of $x$, then

$$\exists \delta>0:B_\delta(x)\subset N$$ , then:

• Take $$y=\text{Max}\left(\frac{1}{2}\delta,\frac{1}{2}\right)$$ , then $$y\in(0,1)$$ and $$y\in N$$ thus $0$ is certainly a limit point, but $0\notin(0,1)$

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TODO: This proof could be nonsense

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See also

• Relatively closed
• Open set
• Neighbourhood