# Convergent nets in a topological space

Let $X$ be a set and $\mathcal{T}_1$, $\mathcal{T}_2$ be two topologies on $X$. Suppose the following is true:

for any net $N=\{x_\alpha\}_{\alpha\in A}$ in $X$, $N$ is convergent in $\mathcal{T}_1$ iff it is convergent in $\mathcal{T}_2$.

Then we can conclude that $\mathcal{T}_1=\mathcal{T}_2$.

I've vaguely heard this statement somewhere before but I'm not sure if it is true. A quick search on Google returns two related MO questions:

I'm reading Introduction to Topology by Gameline and Greene and I'm not familiar with Kelley's big book. Could anyone give some ideas of the proof if the statement is true? (Or we have an counterexample?)

a) If $S$ is a net such that $S_n = s$ for each $n$ [i.e., a constant net], then $S$ converges to $s$.
b) If $S$ converges to $s$, so does each subnet.
c) If $S$ does not converge to $s$, then there is a subnet of $S$, no subnet of which converges to $s$.
d) (Theorem on iterated limits): Let $D$ be a directed set. For each $m \in D$, let $E_m$ be a directed set, let $F$ be the product $D \times \prod_{m \in D} E_m$ and for $(m,f)$ in $F$ let $R(m,f) = (m,f(m))$. If $S(m,n)$ is an element of $X$ for each $m \in D$ and $n \in E_m$ and $\lim_m \lim_n S(m,n) = s$, then $S \circ R$ converges to $s$. He has previously shown that in any topological space, convergence of nets satisfies a) through d). (The first three are easy; part d) is, I believe, an original result of his.) In this section he proves the converse: given a set $S$ and a set $\mathcal{C}$ of pairs (net,point) satisfying the four axioms above, there exists a unique topology on $S$ such that a net $N$ converges to $s \in X$ iff $(N,s) \in \mathcal{C}$.
So now assume every net N that converges in a topology $\tau_1$ on X to a point $x\in X$ also converges to $x\in X$ in a topology $\tau_2$ on X. Now apply the axioms to obtain the proof of the result.