43: Local Criterion for Openness

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of local criterion for openness

---

Topics

About: topological space

---

The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

---

Starting Context

• The reader knows a definition of topological space.

---

Target Context

• The reader will have a description and a proof of the local criterion for openness.

---

Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

---

Main Body

---

1: Structured Description

Here is the rules of Structured Description.

Entities:
$T$: $\in \{\text{ the topological spaces }\}$
$S$: $\subseteq T$
//

Statements:
$S\in \{\text{ the open subsets of }T\}$.
$⟺$
$\mathrm{\forall }p\in S(\mathrm{\exists }{U}_p\subseteq T(U_p\in \{\text{ the open subsets of }T\}\wedge p\in U_p\subseteq S))$.
//

---

2: Natural Language Description

For any topological space, $T$, any subset, $S\subseteq T$, is open if and only if for any point, $p\in S$, there is an open neighborhood, $U_p$, of $p$ such that $p\in U_p\subseteq S$.

---

3: Proof

Let us suppose that $U_p$ exists.

$S$ is the union of such all the $U_p$ s, because any point in $S$ belongs to the union and any point in the union belongs to $S$, so, as an union of open sets, $S$ is an open set.

Let us suppose that $S$ is open.

$S$ is a $U_p$.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>