nLab
convergence

This entry is about the notion of limit in analysis and topology. For the notion of the same name in category theory see at limit.

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

A limit of a sequence (or net) of points (x i)(x_i) in a topological space (or other convergence space) XX is a point xx such that the sequence eventually gets arbitrarily close to xx. We can also speak of a limit of a filter on XX.

The notion is of particular and historical importance in analysis, where it serves to define for instance the notion of derivative.

Definitions

The precise definition depends on what sort of space XX is.

Definition

If XX a topological space and II the set of natural numbers (or more generally any directed set) and ν=(x i) iI:IX\nu = (x_i)_{i\in I}\colon I \to X is a sequence (or a net) of points in XX, one says that a point xXx \in X is a limit of ν\nu or that ν\nu converges to xx if for each neighbourhood UU in XX of xx there exists an nIn \in I such that x iUx_i \in U for each ini \geq n.

An important special case (the original) is:

Definition

If XX the real line and II the set of natural numbers (or more generally any directed set) and ν=(x i) iI:IX\nu = (x_i)_{i\in I}\colon I \to X is a sequence (or a net) of real numbers, one says that a point xXx \in X is a limit of ν\nu or that ν\nu converges to xx if for each positive number ϵ\epsilon there exists an nIn \in I such that |x ix|<ϵ{|x_i - x|} \lt \epsilon for each ini \geq n.

An important generalization (possibly the most general) is:

Definition

If XX a convergence space and II the set of natural numbers (or more generally any directed set) and ν=(x i) iI:IX\nu = (x_i)_{i\in I}\colon I \to X is a sequence (or a net) of points in XX, one says that a point xXx \in X is a limit of ν\nu or that ν\nu converges to xx if the eventuality filter of ν\nu converges to xx (which is a primitive concept in convergence spaces).

Other types of space for which we might put in definitions (or which might have definitions on their own pages) are (extended) (quasi)-(pseudo)-metric spaces, (quasi)-uniform spaces, pretopological spaces, and (quasi)-uniform convergence spaces.

Notation

When one of the conditions above holds, we may write any of the following, where ‘\to’ is read as ‘converges to’:

  • νx\nu \to x,
  • x i ixx_i \to_i x,
  • x ixx_i \to x (suppressing the index ii by abuse of notation).

Or we may write any of the following, were lim\lim is read as ‘the set of limits of’:

  • xlimνx \in \lim \nu,
  • xlim ix ix \in \lim_i x_i,
  • xlimx ix \in \lim x_i (suppressing the index ii by abuse of notation).

Of course, the right-hand side has a meaning by itself, as the set of limits itself (a subset of the underlying set of XX, or a subspace of XX itself).

If XX is a Hausdorff space, then there is at most one point xx with the property that the sequence (or net) ν\nu converges to xx. Then we may write any of the following, were now lim\lim is read as ‘the limit of’:

  • x=limνx = \lim \nu,
  • x=lim ix ix = \lim_i x_i,
  • x=limx ix = \lim x_i (suppressing the index ii by abuse of notation).

Now the right-hand side by itself is the possibly undefined term for the limit itself (if it exists).

Limits of filters

More generally than sequences, and equivalently to nets, we may speak of limits of filters on XX. This concept is axiomatized directly in the concept of convergence space. In the case of a topological space XX, a filter of subsets of XX converges to a point xx if every neighbourhood of xx is contained in the filter.

In the definitions above, equivalent nets (those with equal eventuality filters) always converge to the same point. As every proper filter is the eventuality filter of some net, a proper filter converges to xx if any of these nets converges to xx; the improper filter converges to every point. (In constructive mathematics, we may cover all filters by saying: FF converges to xx if, on the assumption that FF is proper, any of its nets converges to xx.)

Properties

Relation to limits in the sense of category theory

The limits of category theory are a great generalization of an analogy with the limits discussed here. It turns out, however, that limits in topological spaces (at least) can be viewed as category-theoretic limits. For now, see this math.sx answer.

Examples

References

Discussion of this history of the concept, with emphasis on its roots all the way back in Zeno's paradoxes of motion is in

  • Carl Benjamin Boyer, The history of the Calculus and its conceptual development, Dover 1949

category: analysis

Revised on September 19, 2017 08:48:11 by Urs Schreiber (185.25.95.132)