Search results
Results From The WOW.Com Content Network
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.More explicitly, a topological space is second-countable if there exists some countable collection = {} = of open subsets of such that any open subset of can be written as a union of elements of some subfamily of .
Split interval, also called the Alexandrov double arrow space and the two arrows space − All compact separable ordered spaces are order-isomorphic to a subset of the split interval. It is compact Hausdorff, hereditarily Lindelöf, and hereditarily separable but not metrizable. Its metrizable subspaces are all countable. Specialization (pre)order
second-countable space: the topology has a countable base; separable space: there exists a countable dense subspace; Lindelöf space: every open cover has a countable subcover; σ-compact space: there exists a countable cover by compact spaces; Relations: Every first countable space is sequential. Every second-countable space is first-countable ...
Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf. To further compare these two properties: An arbitrary subspace of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set; first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base; separable space: there exists a countable dense subset
Lindelöf's lemma is also known as the statement that every open cover in a second-countable space has a countable subcover (Kelley 1955:49). This means that every second-countable space is also a Lindelöf space.
A Lindelöf space is compact if and only if it is countably compact. Every second-countable space is Lindelöf, [5] but not conversely. For example, there are many compact spaces that are not second-countable. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable. [6] Every regular Lindelöf space ...
In many formulations of the Hahn–Mazurkiewicz theorem, second-countable is replaced by metrizable. These two formulations are equivalent. In one direction a compact Hausdorff space is a normal space and, by the Urysohn metrization theorem, second-countable then implies metrizable. Conversely, a compact metric space is second-countable.