Countability axioms in topology: Difference between revisions
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'''Countability axioms in topology''' are properties that a [[topological space]] may satisfy which refer to the [[countable set|countability]] of certain structures within the space. | '''Countability axioms in topology''' are properties that a [[topological space]] may satisfy which refer to the [[countable set|countability]] of certain structures within the space. | ||
Latest revision as of 01:19, 18 February 2009
Countability axioms in topology are properties that a topological space may satisfy which refer to the countability of certain structures within the space.
A separable space is one which has a countable dense subset.
A first countable space is one for which there is a countable filter base for the neighbourhood filter at any point.
A second countable space is one for which there is a countable base for the topology.
The second axiom of countability implies separability and the first axiom of countability. Neither implication is reversible.