Countable Compactness is Preserved under Continuous Surjection

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces.

Let $\phi: T_A \to T_B$ be a continuous surjection.


If $T_A$ is countably compact, then $T_B$ is also countably compact.


Proof

Let $T_A$ be countably compact.

Take a countable open cover $\UU$ of $T_B$.

From Preimage of Cover is Cover, $\VV := \set {\phi^{-1} \sqbrk U: U \in \UU}$ is a cover of $S_A$.

$\VV$ is a countable cover because there it is bijective with $\UU$.


This article, or a section of it, needs explaining.
In particular: It needs to be demonstrated that "it" is bijective, and in this context, exactly what "it" is.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


By hypothesis, $\phi$ is continuous.

For all $U \in \UU$, $U$ is open in $T_B$.

It follows that $\forall U \in \UU: \phi^{-1} \sqbrk U$ is open in $T_A$.

So $\VV$ is a countable open cover of $T_A$.

$T_A$ is countably compact, so we take a finite subcover:

$\set {\phi^{-1} \sqbrk {U_1}, \ldots, \phi^{-1} \sqbrk {U_n} }$

We have that $\phi$ is surjective.

So from Surjection iff Right Inverse:

$\phi \sqbrk {\phi^{-1} \sqbrk A} = A$

So:

$\set {\phi \sqbrk {\phi^{-1} \sqbrk {U_1} }, \ldots, \phi \sqbrk {\phi^{-1} \sqbrk {U_n} } } = \set {U_1, \ldots, U_n} \subseteq \UU$

is a finite subcover of $\UU$ on $T_B$.


This article, or a section of it, needs explaining.
In particular: It is not clear that $\set {U_1, \ldots, U_n}$ covers $T_B$.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Countable_Compactness_is_Preserved_under_Continuous_Surjection&oldid=469517"