Notes:Generalising the limit

Overview

I can parrot you back a few definitions for limit. These all have something similar though, it's just hard to phrase! You're dragging through the space with a 'sequence' of some sort and seeing what is left as you sweep along it. I want to distil this into a concrete definition.

Primitives

The most important primitive is what I call the net. This is like a net in fishing, you drag it through the water and see what is left in it. In the same way you sweep this along the "space" and see what you end up with it.

This applies to topological spaces (of which the power set is - so it has a use in sets), there was no point in attempting further abstraction as "converges in a group" makes little sense, and using it with measures would at most involve turning the measure into a metric, and I already have limits in metric spaces.

Net

Given any topological space, [ilmath](X,\mathcal{ J })[/ilmath] a net is a tuple consisting of a poset with the forward property, [ilmath](P,\preceq)[/ilmath] and a map, [ilmath]f:P\rightarrow X[/ilmath] where:

• [ilmath](P,\preceq)[/ilmath] is a poset with an additional property:
  • The forward property: [ilmath]\forall u,v\in P\exists w\in P[w\succeq u\wedge w\succeq v][/ilmath], that is:
    • Given any two elements of [ilmath]P[/ilmath] there exists something that is "infront of them both"
• [ilmath]f:P\rightarrow X[/ilmath] is just a map, there are no extra conditions
  • For an understanding of the map, think of a sequence, as mentioned on the sequence page, a sequence is really just a map; [ilmath]f:\mathbb{N}\rightarrow X[/ilmath]. It is easy to see [ilmath]\mathbb{N} [/ilmath] satisfies the forward property.

Here [ilmath]((P,\preceq),f:P\rightarrow X)[/ilmath] is a net, or [ilmath](P,f)[/ilmath] for short.

Captures

Let [ilmath]A\in\mathcal{P}(X)[/ilmath] be any subset of topological space, [ilmath](X,\mathcal{ J })[/ilmath]. Let [ilmath](P,f)[/ilmath] be a net as defined above. We say "the net captures [ilmath]A[/ilmath]" if:

• [ilmath]\exists u\in P\ \forall v\in P[u\preceq v\implies f(v)\in A][/ilmath]

Collides

Let [ilmath]A\in\mathcal{P}(X)[/ilmath] be any subset of topological space, [ilmath](X,\mathcal{ J })[/ilmath]. Let [ilmath](P,f)[/ilmath] be a net as defined above. We say "the net collides with [ilmath]A[/ilmath]" if:

• [ilmath]\forall u\in P\ \exists v\in P[u\preceq v\implies f(v)\in A][/ilmath]

(Terminology note: keeping with the idea of a net as a plane that will block certain things in stream of subsets flowing towards the net in a stream perfectly perpendicular to the surface, but sufficiently turbulent as to not let things balance on the net, a subset collides with the net, then wobbles and passes through, and is not captured by it)

Subnet

We can think of a sub-sequence as a sequence combined with a monotonic sequence [ilmath]\alpha:\mathbb{N}\rightarrow\mathbb{N} [/ilmath] to create a new sequence that picks out (going forward) terms of the initial sequence. Monotonicity is slightly too strong, instead we can get away with:

• For posets [ilmath](P,\preceq)[/ilmath] and [ilmath](Q,\le)[/ilmath] - both satisfying the forward property - and a function [ilmath]\alpha:Q\rightarrow P[/ilmath] if[ilmath]\alpha[/ilmath] satisfies:
  • [ilmath]\forall u\in P\ \exists x\in Q\ \forall y\in Q[x\le y\implies u\le \alpha(y)][/ilmath] - Caution:I am still developing this idea, I think this works but I have not completed my work. The obvious tests are that composition preserves "the forwardness property" and that it works for a sub-sequence, on the page for subsequences (permalink) I note that it must be strictly increasing, not just increasing; I'm also a little bit proud that I'm talking of some "forwardness" property I have now made formal. TL;DR: Pretty sure this works, your mileage may vary.

Then a subnet of [ilmath](P,f)[/ilmath] is simply another net [ilmath](Q,f\circ\alpha)[/ilmath] where [ilmath]\alpha[/ilmath] is a "forwardness preserving function"

• This definition is actually useful, not only should it exist because we have sub-sequences (so a sequence considered as a net should have subnets for its subsequences) but also in claim 4, a generalisation of the Bolzano-Weierstrass theorem.

Convergence

We are now in a position to state convergence:

• A net [ilmath](P,f)[/ilmath] is said to converge to [ilmath]x\in X[/ilmath] if:
  • For all neighbourhoods to [ilmath]x[/ilmath] the neighbourhood is captured by the net.

Caveat

1. In a non-Hausdorff space the limit need not be unique. Take the net [ilmath]f:\mathbb{N}\rightarrow X[/ilmath] with [ilmath]f:(\text{even }n)\mapsto x_1[/ilmath] and [ilmath]f:(\text{odd }n)\mapsto x_2[/ilmath] where [ilmath]x_1[/ilmath] and [ilmath]x_2[/ilmath] are in the same open set and cannot be separated by intersection with other open sets, then:
   • [ilmath](\mathbb{N},f)[/ilmath] only captures neighbourhoods which contain both [ilmath]x_1[/ilmath] and [ilmath]x_2[/ilmath]
     • Thus by our definition, [ilmath](\mathbb{N},f)[/ilmath] will converge to both [ilmath]x_1[/ilmath] and [ilmath]x_2[/ilmath]. (By hypothesis no open set contains just one of [ilmath]x_1[/ilmath] or [ilmath]x_2[/ilmath], it must contain both, so any (all) neighbourhoods to either [ilmath]x_1[/ilmath] or [ilmath]x_2[/ilmath] will contain the other)

Claims

1. In a Hausdorff space a net converges to to a unique point, if it converges at all ("the limit" is now established) - PROVED (uses claim 5)
2. [ilmath]g:X\rightarrow Y[/ilmath] is continuous at a point [ilmath]x\in X[/ilmath] [ilmath]\iff[/ilmath] for every net [ilmath](P,f)[/ilmath] converging to [ilmath]x[/ilmath] the net [ilmath](P,g\circ f)[/ilmath] converges to [ilmath]g(x)[/ilmath] in [ilmath]Y[/ilmath] - PROVED
3. [ilmath]g:X\rightarrow Y[/ilmath] is continuous [ilmath]\iff[/ilmath] for all [ilmath]x\in X[/ilmath] ([ilmath]g[/ilmath] is continuous at [ilmath]x[/ilmath] in the net sense) - PROVED (corollary)
   • We already know that continuous at a point in the net sense is equivalent to the usual continuous at a point and we know continuous at a every point is equivalent to continuous.
4. Analogue of the Bolzano-Weirstrauss theorem: every bounded sequence has a convergent subsequence - A net [ilmath](P,f)[/ilmath] which collides with every neighbourhood of [ilmath]x\in X[/ilmath] has a subnet which converges to [ilmath]x[/ilmath] - PROVED (requires claim 5)
5. If a net captures [ilmath]U[/ilmath] and [ilmath]V[/ilmath] then it captures [ilmath]U\cap V[/ilmath] - PROVED - remember that a topology is closed under finite intersection so this is applicable.
   • note that if [ilmath]U\cap V=\emptyset[/ilmath] that it is impossible for the net to capture both!