Can we characterize a discrete topological space $X_\mathcal d $ by the convergence of nets in it, if yes then how ?
l know that any convergent net in $X_d$ is eventually constant, but does the converse hold ? , i.e if in a space any convergent net is eventually constant then the space is discrete. I came to know that in case of sequences the result does not hold from here.
Any insight about the characterization.Thank you.
Yes, in every non-discrete space there is a convergent net which is not eventually constant.
Suppose that $X$ is a non-discrete space. Then there is a point $x \in X$ which is not isolated, meaning that every open neighborhood of $x$ contains some point distinct from $x$. Now, either $x$ has a smallest open neighborhood, or it doesn't.
If $x$ has a smallest open neighborhood, $U$, then pick $y \in U \setminus \{ x \}$. Consider the sequence $\{ x_n \}_{n \in \mathbb{N}}$ defined by $$x_n = \begin{cases} x, &\text{if $n$ is even}\\ y, &\text{if $n$ is odd.} \end{cases}$$ This sequence is not eventually constant, and it converges to $x$.
If $x$ does not have a smallest open neighborhood, let $\mathcal{N}$ be the family of all open neighborhoods of $x$, and for each $U \in \mathcal{N}$ choose some $x_U \in U \setminus \{ x \}$.
Set $D = \mathcal{N} \times \{ 0,1 \}$ and consider the relation $\preceq$ on $D$ defined by $$( U,i ) \preceq ( V,j ) \Leftrightarrow U \supseteq V\text{, or } U = V \mathrel{\&} i \leq j.$$ Note that $D$ is directed by $\preceq$. Define the net $\{ x_{(U,i)} \}_{(U,i) \in D}$ by $$x_{(U,i)} = \begin{cases} x_U, &\text{if $i=0$} \\ x, &\text{if $i=1$.} \end{cases}$$
This net is not eventually constant, and it converges to $x$.