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Let $X$ be a set. A net is a map from a directed set to $X$. In other words, it is a pair $(A,\gamma)$ where $A$ is a directed set and $\gamma$ is a map from $A$ to $X$. If $a\in A$ then $\gamma(a)$ is normally written $x_{a}$, and then the net is written $(x_{a})_{{a\in A}}$, or simply $(x_{a})$ if the direct set $A$ is understood.
Now suppose $X$ is a topological space, $A$ is a directed set, and $(x_{a})_{{a\in A}}$ is a net. Let $x\in X$. Then $(x_{a})$ is said to converge to $x$ if whenever $U$ is an open neighbourhood of $x$, there is some $b\in A$ such that $x_{a}\in U$ whenever $a\geq b$.
Similarly, $x$ is said to be an accumulation point (or cluster point) of $(x_{a})$ if whenever $U$ is an open neighbourhood of $x$ and $b\in A$ there is $a\in A$ such that $a\geq b$ and $x_{a}\in U$.
Nets are sometimes called Moore–Smith sequences, in which case convergence of nets may be called Moore–Smith convergence.
If $B$ is another directed set, and $\delta\colon B\rightarrow A$ is an increasing map such that $\delta(B)$ is cofinal in $A$, then the pair $(B,\gamma\circ\delta)$ is said to be a subnet of $(A,\gamma)$. Alternatively, a subnet of a net $(x_{\alpha})_{{\alpha\in A}}$ is sometimes defined to be a net $(x_{{\alpha_{\beta}}})_{{\beta\in B}}$ such that for each $\alpha_{0}\in A$ there exists a $\beta_{0}\in B$ such that $\alpha_{\beta}\geq\alpha_{0}$ for all $\beta\geq\beta_{0}$.
Nets are a generalisation of sequences, and in many respects they work better in arbitrary topological spaces than sequences do. For example:

If $Y$ is a subspace of $X$ then $x\in\overline{Y}$ if and only if there is a net in $Y$ converging to $x$.

if $X^{{\prime}}$ is another topological space and $f\colon X\rightarrow X^{{\prime}}$ is a map, then $f$ is continuous at $x$ if and only if whenever $(x_{a})$ is a net converging to $x$, $(f(x_{a}))$ is a net converging to $f(x)$.

$X$ is compact if and only if every net has a convergent subnet.
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Attached Articles
topology via converging nets by CWoo
nets and closures of subspaces by azdbacks4234
every net has a universal subnet by asteroid
accumulation points and convergent subnets by azdbacks4234
compactness and accumulation points of nets by azdbacks4234
continuity and convergent nets by azdbacks4234
topological properties and nets by asteroid
Comments
ordering of the net
Hi, I am new to this forum and am not in particular very good in set theory, but I could not quite comprehend the claim made in the first paragraph of the definition for "net", that xA<=aB iff A<=B. Doesnt the mapping have to be monotonous for that claim to be valid?
Regards,
Dinesh
Re: ordering of the net
> ... I could not quite comprehend the
> claim made in the first paragraph of the definition for
> "net", that xA<=aB iff A<=B. Doesnt the mapping have to be
> monotonous for that claim to be valid?
The statement in question is a definition. The defined order (on X) is "induced" by \gamma and the order on A. Perhaps the difficulty is that unnecessary confusion is introduced by using \leq ambiguously to denote distinct orders.
Unless you are saying more than I think you are, "monotone" works better than "monotonous".
Re: ordering of the net
thanks, that helped.
Re: ordering of the net
I think the claim is wrong. Consider x_n = (1)^n. (Always a good example to try!) Then this sequence is a net defined on a directed set. Since 3 =< 4, the order you mention should have 1 = (1)^3 =< (1)^4 = 1, yet, in a similar way, taking 4 =< 5, arrive at 1 =< 1, and if this were really a partial order (let alone a directed set!) antisymmetry gives 1 = 1, a contradiction.