383: Open Set on Open Topological Subspace Is Open on Base Space

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

A description/proof of that open set on open topological subspace is open on base space

---

Topics

About: topological space

---

The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

---

Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of subspace topology.

---

Target Context

• The reader will have a description and a proof of the proposition that any open set on any open topological subspace is open on the base space.

---

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: Description

For any topological space, $T_1$, and any open topological subspace, $T_2\subseteq T_1$, of $T_1$, any open set, $U\subseteq T_2$, on $T_2$, is open on $T_1$.

---

2: Proof

By the definition of subspace topology, $U=U^{\prime }\cap T_2$ where $U^{\prime }\subseteq T_1$ is open on $T_1$. As $T_2$ is open on $T_1$, $U$ is open on $T_1$ as an intersection of finite open sets.

---

References

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