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A first-countable, separable Hausdorff space (in particular, a separable metric space) has at most the continuum cardinality. In such a space, closure is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset ...
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 .
Separable. A space is separable if it has a countable dense subset. First-countable. A space is first-countable if every point has a countable local base. Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf. Lindelöf.
is a Hausdorff space if any two distinct points in are separated by neighbourhoods. This condition is the third separation axiom (after T 0 and T 1), which is why Hausdorff spaces are also called T 2 spaces. The name separated space is also used. A related, but weaker, notion is that of a preregular space.
A topological space with a countable dense subset is called separable. A topological space is a Baire space if and only if the intersection of countably many dense open sets is always dense. A topological space is called resolvable if it is the union of two disjoint dense subsets.
Every second-countable space is separable. A metric space is separable if and only if it is second-countable and if and only if it is Lindelöf. Clearly a MS is a space so if separable iff second countable; so should the second one not be In a Metric space the following are equivalent: -- space 2nd countable -- space separable -- space Lindelof
The existence of a line separating the two types of points means that the data is linearly separable In Euclidean geometry , linear separability is a property of two sets of points . This is most easily visualized in two dimensions (the Euclidean plane ) by thinking of one set of points as being colored blue and the other set of points as being ...
A topological space X is connected if these are the only two possibilities. Conversely, if a nonempty subset A is separated from its own complement, and if the only subset of A to share this property is the empty set, then A is an open-connected component of X .