Equality of two topological spaces

Question

Suppose $(X, T_1), (X, T_2)$ be two topological spaces. It is given that a sequence $\{x_n\}$ is convergent in $T_1$ if and only if it is convergent in $T_2.$ Does this imply $T_1= T_2\;$ ? If not, I can't find any counterexamples.

Answer 1

The answer is negative. Consider two topologies $\tau_1$ and $\tau_2$ in $\mathbb R$:

• $\tau_1$ is the discrete topology;
• $\tau_2$ is the topology for which a set $S\neq\mathbb R$ is closed if and only if $S$ is finite or countable.

Obviously, $\tau_1\neq\tau_2$. Now, prove that the convergent sequences are the same in $(\mathbb{R},\tau_1)$ and in $(\mathbb{R},\tau_2)$.

Answer 2

consider $\omega_1+1$ with the order topology $\tau$, and the topology generated by $\tau\cup\{\{\omega_1\}\}$.