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 $A$ of any topological space, a point $x$ belongs to the closure of $A$ if and only if $x$ is a limit point of at least one net whose elements belong to $A$.
A topological space is Frechet–Uryson (or Frechet–Urysohn) if a point $x$ of the closure of any given subset $A$ of $X$ is a limit point of at least one sequence whose elements belong to $A$.
second-countable: there is a countable base of the topology.
metrisable: the topology is induced by a metric.
$\sigma$-locally discrete base: the topology of $X$ is generated by a $\sigma$-locally discrete base.
$\sigma$-locally finite base: the topology of $X$ 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.
countable choice: the natural numbers is a projective object in Set.
metacompact: every open cover has a point-finite open refinement.
first-countable: every point has a countable neighborhood base
Frechet-Uryson space: the closure of a set $A$ consists precisely of all limit points of sequences in $A$
sequential topological space: a set $A$ is closed if it contains all limit points of sequences in $A$
countably tight: for each subset $A$ and each point $x\in \overline A$ there is a countable subset $D\subseteq A$ such that $x\in \overline D$.
a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of countable cover of $X$.
second-countable spaces are separable: use the axiom of countable choice to choose a point in each set of countable cover.
weakly Lindelöf spaces with countably locally finite base are second countable.
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 $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-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.
Last revised on April 5, 2019 at 19:48:15. See the history of this page for a list of all contributions to it.