First-Countability is not Continuous Invariant

Theorem

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

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

If $T_A$ is a first-countable space, then it does not necessarily follow that $T_B$ is also first-countable.

Proof

Let $T_S = \struct {S, \tau_S}$ be the Arens-Fort space.

Let $T_D = \struct {S, \tau_D}$ be the discrete space, also on $S$.

Let $I_S: S \to S$ be the identity mapping on $S$.

From Mapping from Discrete Space is Continuous, we have that $I_S$ is a continuous mapping.

Then we have:

Discrete Space is First-Countable

Arens-Fort Space is not First-Countable

Thus we have demonstrated a continuous mapping from a first-countable space to a space which is not first-countable space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space: $6$