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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.

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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 $x$.

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  • $\begingroup$ So we can characterize the discrete topological space by convergent net , right ? $\endgroup$ Jul 30 '17 at 4:20
  • $\begingroup$ One more thing, I am okey with yout proof, but can't we prove it directly,i.e if every convergent net is eventually constant,then the space is discrete ? $\endgroup$ Jul 30 '17 at 4:22
  • $\begingroup$ @HirenGarai This proof shows " if every convergent net is eventually constant then $X$ is discrete" via the contrapositive. So it has already been shown. It's a completely direct proof too. What's wrong with a proof by contrapositive ? $\endgroup$ Jul 30 '17 at 6:22
  • $\begingroup$ I have no problem with this proof, actually I am a novice in Topology and as usual I was trying to prove the result by assuming that every convergent sequence is eventually constant.So I have to show that every singleton set is open and this is where I get stuck so that's why I was looking for a direct proof..@HennoBrandsma $\endgroup$ Jul 30 '17 at 7:00
  • $\begingroup$ @HirenGarai I cannot think of how one would start with the assumption that all convergent nets are eventually constant and then show that the space is discrete. Perhaps it can be done and I just don't have a strong enough intuition for that hypothesis. Since I do have a stronger feeling for the hypothesis that a space is not discrete, I chose to work via contrapositive. $\endgroup$ Jul 30 '17 at 7:05

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