Sigma-Compactness is Preserved under Continuous Surjection

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 $\sigma$-compact, then $T_B$ is also $\sigma$-compact.

Proof

Let $T_A$ be $\sigma$-compact.

Then:

$\displaystyle S_A = \bigcup_{i \mathop = 1}^\infty S_i$

where $S_i \subseteq S_A$ are compact.

Since $\phi$ is surjective, we have from Image of Union under Relation:

$\displaystyle \phi \sqbrk {S_A} = S_B = \phi \sqbrk {\bigcup_{i \mathop = 1}^\infty S_i} = \bigcup_{i \mathop = 1}^\infty \phi \sqbrk {S_i}$

From Compactness is Preserved under Continuous Surjection, we have that $\phi \sqbrk {S_i}$ is compact for all $i \in \N$.

So $S_B$ is the union of a countable number of compact subsets.

Thus, by definition, $T_B$ is also $\sigma$-compact.

$\blacksquare$