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:

*

*Is a topology determined by its convergent sequences?

*How do you axiomatize topology via nets?
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: Ok,a great partial answer to this question was given by Pete Clark at Math Overflow in the second link you posted. I'd strongly recommend you read his wonderful notes on general convergence at his website,you'll also find the link in his post. But proving your claim as a result of the Kelley axioms is a very good and useful exercise for you to attempt yourself. Recall again the Kelley axioms quoted from Pete's post: 
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. 
It's not a simple proof,but it's worth the effort.  
