nLab
sequential topological space

Contents

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 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 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.)

propertiesimplications
metrisable: topology is induced by a metrica metric space has a σ \sigma -locally discrete base
second-countable: there is a countable base of the topology.Nagata-Smirnov metrization theorem
σ\sigma-locally discrete base, i.e. XX has a σ \sigma -locally discrete base.a second-countable space has a σ \sigma -locally finite base: take the the collection of singeltons of all elements of countable cover of XX.
σ\sigma-locally finite base, i.e. XX has a countably locally finite base.second-countable spaces are separable: choose a point in each set of countable cover.
separable: there is a countable dense subset.second-countable spaces are Lindelöf
Lindelöf: every open cover has a countable sub-cover.weakly Lindelöf spaces with countably locally finite base are second countable
weakly Lindelöf: every open cover has a countable subcollection the union of which is dense.separable metacompact spaces are Lindelöf
countable choice: the natural numbers is a projective object in Set.separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover.
metacompact: every open cover has a point-finite open refinement.Lindelöf spaces are trivially also weakly Lindelöf.
first-countable: every point has a countable neighborhood basea space with a σ\sigma-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a σ\sigma-locally finite base.
Frechet-Uryson space: the closure of a set AA consists precisely of all limit points of sequences in AAa first-countable space is Fréchet-Urysohn: obvious
sequential topological space: a set AA is closed if it contains all limit points of sequences in AAobviously, a Fréchet-Uryson space is sequential
countably tight: for each subset AA and each point xA¯x\in \overline A there is a countable subset DAD\subseteq A such that xD¯x\in \overline D.obviously, a sequential space is countably tight.

References

  • R. Engelking, General topology

Last revised on April 5, 2019 at 19:42:32. See the history of this page for a list of all contributions to it.