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 .
Let be a countable basis of .Consider an open cover, =.To get prepared for the following deduction, we define two sets for convenience, := {:}, ′:=. A straight-forward but essential observation is that, = which is from the definition of base. [1]
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 ...
Every second-countable space is first-countable, separable, and Lindelöf. Semilocally simply connected A space X is semilocally simply connected if, for every point x in X, there is a neighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x. Every simply connected space and every locally simply connected ...
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 ...
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
Main page; Contents; Current events; Random article; About Wikipedia; Contact us; Pages for logged out editors learn more
The real numbers form a topological group under addition. In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.