A net vs a sequence

Question

A net is defined as a map $\Theta\to \mathbb{X}$ ($\theta\mapsto x_{\theta}$) where $\Theta$ is a directed set and $\mathbb{X}$ is some topological space. If $\Theta=\mathbb{N}$ then this definition coincides with the usual definition of a sequence. What would be an example of a net which is not a sequence? In particular for sequences we have that two different indices say $k\neq j$ map to the same element $x\in \mathbb{X}$ i.e. $x_k=x_j$. But the same index cannot be mapped to two different elements in $\mathbb{X}$ i.e. we cannot have $x_k=y_k$ where $x$ and $y$ are different. Is this the criteria which differentiates sequences from nets in general? Or do I get all this wrong?

Answer 1

An example that's important in proving that nets do the good things that they do: Say $X$ is a topological space and $p\in X$. Let $\Theta$ be the collection of all neighborhoods of $p$, ordered by reverse inclusion (so $V\ge W$ if $V\subset W$.

Recall that if $(x_\alpha)$ is a net in $X$ we say that $x_\alpha\to x$ if for every neighborhood $V$ of $x$ there exists $\beta$ such that $x_\alpha\in V$ for all $\alpha\ge\beta$. One of the reasons nets are useful is this:

Theorem Suppose that $X$ and $Y$ are topological spaces and $f:X\to Y$. Then $f$ is continuous at $x\in X$ if and only if $f(x_\alpha)\to f(x)$ for every net $(x_\alpha)\subset X$ with $x_\alpha\to x$.

If $f$ is continuous at $x$ and $x_\alpha\to x$ then it's trivial from the definitions that $f(x_\alpha)\to f(x)$. For the converse we need nets defined using that funny ordered set above.

Say $f$ is not continuous at $x$. So there exists $U\subset Y$ open with $f(x)\in U$ such that $f^{-1}(U)$ does not contain a neighborhood of $x$. Say $\Theta$ is the set of all neighborhoods of $x$, ordered by reverse inclusion. For every $V\in\Theta$ there exists $x_V\in V$ with $f(x_V)\notin U$. So $(x_V)$ is a net in $X$, and it's easy to verify that $x_V\to x$ but $f(x_V)\not\to f(x)$.

(The reason $\Theta$ was ordered by reverse inclusion was so we could show that $x_V\to x$: Say $W$ is a neighborhood of $x$. If $V\ge W$ then $x_V\in V\subset W$, hence $x_V\in W$ for every $V\ge W$.)

Answer 2

Question: What would be an example of a net which is not a sequence?

Answer: The identity map $id:[0,1] \to [0,1]$ is a net that is not a sequence.

EDIT:

Definitions:

A net is a function from a directed set $J$ to a topological space $X.$

A sequence is a function from $\mathbb{N}$ to a topological space $X.$

Consequently, any net where $J$ is not $\mathbb{N}$ is not a sequence.

Answer 3

You already have two examples, but here's another that comes up in calculus (but usually isn't mentioned there). Let $[a,b]$ be some interval of real numbers, and let $J$ be the set of all partitions of $[a,b]$ into finitely many subintervals. Then $J$ is directed set with respect to refinement, i.e., we say that one partition is $\geq$ another if the former is a refinement of the latter. Given any bounded function $f:[a,b]\to\mathbb R$, we can associate to each partition $j\in J$ the upper Darboux sum of $f$ with respect to this partition $j$. This function, from $J$ into $\mathbb R$, is a net. If it converges, then its limit deserves to be called the upper Riemann integral of $f$ over $[a,b]$. (There's another net, with the same directed set, using lower Darboux sums; its limit would be the lower Riemann integral.)