nLab
limit point

Context

Analysis

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

Limit points

Idea

Given a space SS, a subspace AA of SS, and a concrete point xx in SS, xx is a limit point of AA if xx can be approximated by the contents of AA.

There are several variations on this idea, and the term ‘limit point’ itself is ambiguous (sometimes meaning Definition 2, sometimes Definition 3.

Definitions

The classical definitions apply when SS is a topological space. Then AA may be thought of as a subset of (the underlying set of) SS, and xx as an element. In order to apply the definitions in constructive mathematics, there needs to be an inequality relation \ne on the points of SS; in classical mathematics, this is taken to be the denial inequality, as usual. (We need not assume that \ne is an apartness relation nor any compatibility between \ne and the topology, at least for the definitions; although it's quite possible that some classical theorems will require such assumptions.)

For the most general definitions, let κ\kappa be a collection of cardinal numbers. (We might want κ\kappa to have some closure properties akin to those of an arity class, but the definition there is not quite what we want.) Recall that a κ\kappa-ary? indexed subset of SS is a function B:ISB\colon I \to S such that the cardinality of II belongs to the class κ\kappa; a point yy is in BB (as an indexed subset) if yy belongs to the range of BB (as a function), and yy is out of BB if yy is inequal (\ne) to every point in BB.

Definitions

The point xx is a κ\kappa-adherent point of the subspace AA if, for each neighbourhood UU of xx, for each κ\kappa-ary indexed subset BB of the intersection UAU \cap A, there is an element yy of UAU \cap A that is out of BB. Slightly more strongly, xx is a κ\kappa-accumulation point (or κ\kappa-cluster point) of AA if, for each neighbourhood UU of xx, for each κ\kappa-ary indexed subset BB of UAU \cap A, there is an element yxy \ne x of UAU \cap A that is out of BB. (Alternatively, take UU to be a punctured neighborhood?, but that won't work constructively in general.)

Every κ\kappa-accumulation point is a κ\kappa-adherent point; the converse holds if every kκk \in \kappa satisfies k+1κk + 1 \in \kappa (and then one usually says ‘accumulation’ rather than ‘adherent’). Also, if κλ\kappa \subseteq \lambda, then every λ\lambda-(adherent/accumulation) point is a κ\kappa-(adherent/accumulation) point.

It immediately follows that the following classical special cases are in order of increasing strength:

  • Using κ=1={0}\kappa = 1 = \{0\}:

    Definition

    The point xx is an adherent point of the subspace AA if, for every neighbourhood UU of xx, the intersection UAU \cap A is inhabited (nonempty).

  • Using κ=1\kappa = 1 again:

    Definition

    The point xx is an accumulation point of the subspace AA if, for every punctured neighbourhood UU of xx, the intersection UAU \cap A is inhabited.

  • Using κ=ω={0,1,2,}\kappa = \omega = \{0, 1, 2, \ldots\}:

    Definition

    The point xx is an ω\omega-accumulation point of the subspace AA if, for every neighbourhood UU of xx, the intersection UAU \cap A is infinite.

  • Using κ=ω 1={0,1,2,, 0}\kappa = \omega_1 = \{0, 1, 2, \ldots, \aleph_0\}:

    Definition

    The point xx is a condensation point of the subspace AA if, for every neighbourhood UU of xx, the intersection UAU \cap A is uncountable.

Properties

The subspace AA is closed iff every adherent point of AA belongs to AA and iff every accumulation point of AA belongs to AA. (Thus one may say that AA is closed iff every limit point of AA belongs to AA without ambiguity.)

More generally, the closure of AA is the set of all adherent points of AA. Classically (using excluded middle, or more generally if SS has decidable equality), the closure of AA is the union of AA and its set of accumulation points.

Classically, a point in AA that is not an accumulation point of AA is precisely an isolated point?. (Constructively, each of these is stronger than the negation of the other.)

A justification for the terminology ‘limit point’ for an accumulation point is that the concept of limit of a fuction? approaching a point really only makes sense approaching an accumulation point. (This is for essentially the same reason that every function is continuous at an isolated point.) More precisely, every answer whatsoever satisfies the definition of lim cf\lim_c f if cc is a non-accumulation point of domfdom f (because the improper filter converges everywhere).

References

Revised on April 19, 2017 13:20:47 by Urs Schreiber (92.218.150.85)