Open set

(Unknown grade)

This page is currently being refactored (along with many others)

Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.

Definition

Topological space

In a topological space $(X,\mathcal{J})$ we have:

• $\forall S\in\mathcal{J}$ that $S$ is an open set. $\mathcal{J}$ is by definition the set of open sets of $X$

Metric space

In a metric space $(X,d)$ there are 2 definitions of open set, however it will be shown that they are equivalent. Here $U$ is some arbitrary subset of $X$.

I claim that the following definitions are equivalent:

Definition 1

• A set $U\subseteq X$ is open in $(X,d)$ (or just $X$ if the metric is implicit) if $U$ is a neighbourhood to all of its points, that is to say:
  • $U\subseteq X$ is open if $$\forall x\in U\exists\delta_x>0[B_{\delta_x}(X)\subseteq U]$$ - (recall that $B_r(x)$ denotes the open ball of radius $r$ centred at $x$) or
  • For all $x$ in $U$ there is an open ball centred at $x$ entirely contained within $U$

Definition 2

• A set $U\subseteq X$ is open in $(X,d)$ (or just $X$ if the metric is implicit) if:
  • $\text{Int}(U)=U$ - (recall that $\text{Int}(U)$ denotes the interior of $U$), that is to say

    (recall that $\ x\text{ is interior to }U\}$ and that a point, $x$ is interior to $U$ if $\exists\delta>0[B_\delta(x)\subseteq U]$)

It is easy to see that these definitions are very similar to each other (these are indeed equivalent is claim 1)

Immediate results

It is easily seen that:

• $\emptyset$ is open (claim 2)
• $X$ itself is open (claim 3)
• $\text{Int}(U)$ is open

  (see interior for a proof of this. The claim is the same as $\text{Int}(\text{Int}(U))=\text{Int}(U)$ as by the first claim we can use either definition of open, so we use $U$ is open if $U=\text{Int}(U)$)

Proof of claims

[Expand]

Claim 1: A set, $U\subseteq X$ is open according to definition $1$ $\iff$ $U$ is open according to definition $2$

---

---

TODO: Trivial proof, be bothered to do it

---

[Expand]

Claim 2: $\emptyset$ is open

---

---

TODO: Trivial proof, be bothered to do it

---

[Expand]

Claim 3: $X$ itself is open

---

---

TODO: Trivial proof, be bothered to do it

---

See also

• Open ball
• Closed set
• Neighbourhood
• Topological space
• Metric space

Old page

Here

$$(X,d)$$ denotes a metric space, and $$B_r(x)$$ the open ball centred at $$x$$ of radius $$r$$

Metric Space definition

"A set

$$U$$ is open if it is a neighborhood to all of its points" and neighborhood is as you'd expect, "a small area around".

Neighbourhood

A set

$$N$$ is a neighborhood to $$a\in X$$ if $$\exists\delta>0:B_\delta(a)\subset N$$

That is if we can puff up any open ball about $x$ that is entirely contained in $N$

Topology definition

In a topological space the elements of the topology are defined to be open sets

Neighbourhood

A subset $N$ of a Topological space $(X,\mathcal{J})$ is a neighbourhood of $p$ if:

• $$\exists U\in\mathcal{J}:p\in U\wedge U\subset N$$

See also

• Relatively open set
• Closed set
• Relatively closed set