Filtros y redes

Sequences and nets in topology
Stijn Vermeeren (University of Leeds) June 24, 2010

arXiv:1006.4472v1 [math.GN] 23 Jun 2010

In a metric space, such as the real numbers with their standard metric, a set A is open if and only if no sequence with terms outside of A has a limit inside A. Moreover, a metric space is compact if and only if every sequence has a converging subsequence. However, ina general topological space these equivalences may fail. Unfortunately this fact is sometimes overlooked in introductory courses on general topology, leaving many students with misconceptions, e.g. that compactness would always be equal to sequence compactness. The aim of this article is to show how sequences might fail to characterize topological properties such as openness, continuity andcompactness correctly. Moreover, I will define nets and show how they succeed where sequences fail. This article grew out of a discussion I had at the University of Leeds with fellow PhD students Phil Ellison and Naz Miheisi. It also incorporates some work I did while enrolled in a topology module taught by Paul Igodt at the Katholieke Universiteit Leuven in 2010.

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Prerequisites and terminologyI will assume that you are familiar with the basics of topological and metric spaces. Introductory reading can be found in many books, such as [14] and [17]. I will frequently refer to a topological space (X, τ ) by just the unlying set X, when it is irrelevant or clear from the context which topology on X is considered. Remember that any metric space (X, d) has a topology whose basic opens arethe open balls B(x, δ) = {y | d(x, y) < δ} for all x ∈ X and δ > 0. A neighbourhood of a point x in a topological space is an open set U with x ∈ U . Note that some people call U a neighbourhood of x if U just contains an open set containing x [17, p. 97], but in this article neighbourhoods are always open themselves.

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A sequence (xn ) converges to a point y if every neighbourhood of ycontains xn for n large enough. We write xn → y and say that y is a limit of the sequence (xn ). If (xn ) converges to y, then so does every subsequence of (xn ). If f : X → Y is a continuous function and xn → y in X, then also f (xn ) → f (y) in Y . (We say that continuous functions preserve convergence of sequences.) Convergence in a product space is pointwise, i.e. a sequence (xn ) in i∈I Xiconverges to y if and only if xn (i) → y(i) in Xi for all i ∈ I. A topological space is Hausdorff if for every two distinct points x and y, we can find a neighbourhood of x and a neighbourhood of y that are disjoint. Sequences in general can have more than one limit, but in a Hausdorff space limits (if they exist at all) are unique. Indeed, a sequence cannot be eventually in two disjoint neighbourhoods atonce. A set X is countable when there is a surjection from N = {0, 1, 2, 3, . . .} onto X. So X is countable if and only if X is finite of in bijection with the natural numbers. A countable union of countable sets is still countable. Cantor’s famous diagonal argument proves that the unit interval [0, 1] and R are uncountable.[5] A few of my examples will make use of ordinal numbers. If you areunfamiliar with ordinal numbers, you can either find background reading in [12] or you can skip over these examples. I write the first infinite ordinal (i.e. the order type of the natural numbers) as ω0 and the first uncountable ordinal as ω1 . Because a countable union of countable ordinals is still countable, no countable sequence of countable ordinals can have ω1 as limit. In other words: the cofinalityof ω1 is ω1 .

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Open versus sequentially open

In a topological space X, a set A is open if and only if every a ∈ A has a neighbourhood contained in A. A is sequentially open if and only if no sequence in X \ A has a limit in A, i.e. sequences cannot converge out of a sequentially closed set. If X is a metric space, then the two notions of open and sequentially open are equivalent....