A Frechet–Urysohn (or Frechet–Uryson) space is a topological space in which the closure of a subspace may be described using only sequences.
Recall that, given any subset of any topological space, a point belongs to the closure of if and only if is a limit point of at least one net whose elements belong to .
A topological space is Frechet–Uryson (or Frechet–Urysohn) if a point of the closure of any given subset of is a limit point of at least one sequence whose elements belong to .
Axioms: axiom of choice (AC), countable choice (CC).
second-countable: there is a countable base of the topology.
metrisable: the topology is induced by a metric.
-locally discrete base: the topology of is generated by a -locally discrete base.
-locally finite base: the topology of is generated by a countably locally finite base.
Lindelöf: every open cover has a countable sub-cover.
weakly Lindelöf: every open cover has a countable subcollection the union of which is dense.
metacompact: every open cover has a point-finite open refinement.
countable chain condition: A family of pairwise disjoint open subsets is at most countable.
first-countable: every point has a countable neighborhood base
Frechet-Uryson space: the closure of a set consists precisely of all limit points of sequences in
sequential topological space: a set is closed if it contains all limit points of sequences in
countably tight: for each subset and each point there is a countable subset such that .
a second-countable space has a -locally finite base: take the the collection of singeltons of all elements of a countable cover of .
second-countable spaces are separable: use the axiom of countable choice to choose a point in each set of a countable cover.
weakly Lindelöf spaces with countably locally finite base are second countable.
separable spaces satisfy the countable chain condition: given a dense set and a family , the map assigning to the unique with is surjective.
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.
Lindelöf spaces are trivially also weakly Lindelöf.
a space with a -locally finite base is first countable: obviously, every point is contained in at most countably many sets of a -locally finite base.
a first-countable space is obviously Fréchet-Urysohn.
a Fréchet-Uryson space is obviously sequential.
a sequential space is obviously countably tight.
paracompact spaces satisfying the countable chain condition are Lindelöf.
Last revised on April 5, 2019 at 23:48:15. See the history of this page for a list of all contributions to it.