392: 1st-Countable Topological Space Is Sequentially Compact if It Is Countably Compact

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

A description/proof of that 1st-countable topological space is sequentially compact if it is countably compact

---

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 1st-countable topological space.
• The reader knows a definition of sequentially compact topological space.
• The reader knows a definition of countably compact topological space.
• The reader admits the proposition that any topological space is countably compact if and only if each infinite subset has an $\omega$-accumulation point.

---

Target Context

• The reader will have a description and a proof of the proposition that any 1st-countable topological space is sequentially compact if the space is countably compact.

---

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

Any 1st-countable topological space, $T$, is sequentially compact if $T$ is countably compact.

---

2: Proof

Let us suppose that $T$ is countably compact. Let any infinite points sequence be $p_1,p_2,...$. Let us define the subset, $S\subseteq T:=\{p_1,p_2,...\}$. $S$ has an $\omega$-accumulation point, $p\in T$, by the proposition that any topological space is countably compact if and only if each infinite subset has an $\omega$-accumulation point. There is a countable neighborhood basis, $\{B_{p,i}\}$ at $p$. There is a point, $p_1^{\prime }\in B_{p,1}\cap S$. There is a point, $p_2^{\prime }\in B_{p,1}\cap B_{p,2}\cap S$, which can be chosen to be later than $p_1^{\prime }$ in the sequence, because $B_{p,1}\cap B_{p,2}\cap S$ has some infinite points, and so on. After all, there is a point, $p_i^{\prime }\in B_{p,1}\cap B_{p,2}\cap ...\cap B_{p,i}\cap S$, which is later than $p_1^{\prime },p_2^{\prime },...,p_{i-1}^{\prime }$ in the sequence. $p_1^{\prime },p_2^{\prime },...$ is a subsequence of the original sequence. The subsequence converges to $p$, because for any neighborhood, $U_p$, there is a $B_{p,i}\subseteq U_p$, and $p_j^{\prime }\in B_{p,i}\subseteq U_p$ for any $i\le j$.

---

References

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