488: Compact Topological Space

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

definition of compact topological space

---

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

---

Starting Context

• The reader knows a definition of topological space.

---

Target Context

• The reader will have a definition of compact 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: Structured Description

Here is the rules of Structured Description.

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

Conditions:
$\mathrm{\forall }\{U_{\alpha }|\alpha \in A\}$, where $A\in \{\text{ the possibly uncountable index sets }\}$, $U_{\alpha }\in \{\text{ the open subsets of }T\}$, and ${\cup }_{\alpha \in A}{U}_{\alpha }=T$
(
$\mathrm{\exists }J\subseteq A$
(
$J\in \{\text{ the finite sets }\}$
$\wedge$
${\cup }_{j\in J}{U}_j=T$
)
)
//

---

2: Natural Language Description

Any topological space, $T$, such that each open cover of $T$ has a finite subcover

---

References

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