first-countable space

First-countable spaces



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

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


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topological homotopy theory

First-countable spaces


A space (such as a topological space) is first-countable if, in a certain sense, there is only a countable amount of information locally in its topology. (Change ‘locally’ to ‘globally’ to get a second-countable space.)



A topological space is first-countable if every point xx has a countable local basis B xB_x.


The character of a space at a point xx is the minimum of the cardinalities of the possible bases B xB_x. We are implicitly using the axiom of choice here, to suppose that this set of cardinalities (which really is a small set because bounded above by the number of neighbourhoods of xx, and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.

Then a first-countable space is simply one whose characters are all countable.

The character, tout court, of a space is the supremum of the characters of its points; then a first-countable space is simply one with a countable character.

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.

Last revised on April 5, 2019 at 19:49:17. See the history of this page for a list of all contributions to it.