nLab
sequential topological space

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Contents

Idea

A topological space is sequential if (in a certain sense) you can do topology in it using only sequences instead of more general nets.

Sequential spaces are a kind of nice topological space.

Definition

A sequential topological space is a topological space XX such that a subset AA of XX is closed if (hence iff) it contains all the limit points of all sequences of points of AA—or equivalently, such that AA is open if (hence iff) any sequence converging to a point of AA must eventually be in AA.

Equivalently, a topological space is sequential iff it is a quotient space (in TopTop) of a metric space.

Examples

  • Every Frechet–Uryson space is a sequential space.

  • Every topological space satisfying the first countability axiom is Frechet–Uryson, hence a sequential space. In particular, this includes any metrizable space .

  • Every quotient of a sequential space is sequential. In particular, every CW complex is also a sequential space. (Conversely, every sequential space is a quotient of a metrizable space, giving the alternative definition).

Properties

In constructive mathematics

Everything above assumes excluded middle (and probably at least countable choice). Without that, it's hard to prove the existence of any nontrivial sequential spaces.

For example, to prove that the real line is sequential as a topological space, we must find, given a set AA and a point xx such that every sequence converging to xx is eventually in AA, a positive real number δ\delta such that ]xδ,x+δ[A{]x - \delta, x + \delta[} \subseteq A, and it's not clear how to construct that number from the data at hand. (One might consider various specific sequences that converge to xx, such as (x+1/n) n(x + 1/n)_n and (x1/n) n(x - 1/n)_n, and use them to find upper bounds on δ\delta; but no finite set of sequences will give an entire interval around xx, and proving that an infinite set of sequences that does cover an entire interval has a uniform positive upper bound on δ\delta is very tricky.)

The usual proof that the real line (or any first-countable topological space) is sequential uses excluded middle and countable choice: Supposing that AA is not open, consider xAx \in A such that xInt(A)x \notin Int(A), pick for each δ=1/n\delta = 1/n (or for each of the countably many basic neighbourhoods U nU_n of xx in a general first-countable space) a point y ny_n such that d(x,y n)<1/nd(x,y_n) \lt 1/n (or such that y nU ny_n \in U_n) but y nAy_n \notin A (which must exist since none of these balls/neighbourhoods are contained in AA), note that lim ny n=x\lim_n y_n = x, and get a contradiction.

For this reason, constructive analysis often requires the use of general nets (or filters) in situations where classical analysis can get by with sequences. (It is trivially true, in any topological space, that a set AA is open if every net that converges to an element xx of AA belongs eventually to AA, or equivalently that AA belongs to any filter that converges to xx; you just use the neighbourhood filter of xx.)

References

  • R. Engelking, General topology

Revised on May 17, 2017 07:52:31 by Urs Schreiber (92.218.150.85)