First-countable space

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In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point in there exists a sequence of neighbourhoods of such that for any neighbourhood of there exists an integer with contained in Since every neighborhood of any point contains an open neighborhood of that point the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.

Examples and counterexamples[edit]

The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at with radius for integers form a countable local base at

An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line).

Another counterexample is the ordinal space where is the first uncountable ordinal number. The element is a limit point of the subset even though no sequence of elements in has the element as its limit. In particular, the point in the space does not have a countable local base. Since is the only such point, however, the subspace is first-countable.

The quotient space where the natural numbers on the real line are identified as a single point is not first countable.[1] However, this space has the property that for any subset and every element in the closure of there is a sequence in A converging to A space with this sequence property is sometimes called a Fréchet–Urysohn space.

First-countability is strictly weaker than second-countability. Every second-countable space is first-countable, but any uncountable discrete space is first-countable but not second-countable.

Properties[edit]

One of the most important properties of first-countable spaces is that given a subset a point lies in the closure of if and only if there exists a sequence in which converges to (In other words, every first-countable space is a Fréchet-Urysohn space and thus also a sequential space.) This has consequences for limits and continuity. In particular, if is a function on a first-countable space, then has a limit at the point if and only if for every sequence where for all we have Also, if is a function on a first-countable space, then is continuous if and only if whenever then

In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which are not compact (these are necessarily non-metric spaces). One such space is the ordinal space Every first-countable space is compactly generated.

Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable, although uncountable products need not be.

See also[edit]

References[edit]

  1. ^ (Engelking 1989, Example 1.6.18)

Bibliography[edit]

  • "first axiom of countability", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Engelking, Ryszard (1989). General Topology. Sigma Series in Pure Mathematics, Vol. 6 (Revised and completed ed.). Heldermann Verlag, Berlin. ISBN 3885380064.