487: Compact Subset of Topological Space

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

A definition of compact subset of topological space

---

Topics

About: topological space

---

The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

---

Starting Context

• The reader knows a definition of topological space.

---

Target Context

• The reader will have a definition of compact subset of topological 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: Definition

For any topological space, $T$, any subset, $S\subseteq T$, such that each open cover of $S$ has a finite subcover

---

2: Note

$S$ can be regarded as the topological subspace of $T$, and we can talk about compactness of $S$ as the topological space, and compactness of $S$ as the subset and compactness of $S$ as the topological space are not the same by the definitions (an open cover of $S$ as the subset is not necessarily an open cover of $S$ as the subspace; an open cover of $S$ as the subspace is not necessarily an open cover of $S$ as the subset), but in fact, each of the 2 concepts implies the other, by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.

---

References

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