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 that is not first-countable is the cofinite topology on an uncountable set (such as the real line). More generally, the Zariski topology on an algebraic variety over an uncountable field is not first-countable.

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   that 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 that are not compact (these are necessarily not metrizable 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.