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First-countable Space
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EXAMPLES AND COUNTEREXAMPLES


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 Ball s centered at ''x'' with radius 1/''n'' for integers ''n'' > 0 form a countable local base at ''x''.

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 {Link without Title} where ω1 is the smallest Uncountable Ordinal Number . The element ω1 is a Limit Point of the subset [0,ω1) even though no sequence of elements in [0,ω1) has the element ω1 as its limit. In particular, ω1 does not have a countable local base. The subspace [0,ω1) is first-countable however, since ω1 is the only such point.


PROPERTIES


One of the most important properties of first-countable spaces is that given a subset ''A'', a point ''x'' lies in the Closure of ''A'' if and only if there exists a Sequence {''x''''n''} in ''A'' which Converges to ''x''. This has consequences for Limits and Continuity . In particular, if ''f'' is a function on a first-countable space, then ''f'' has a limit ''L'' at the point ''x'' if and only if for every sequence ''x''''n'' → ''x'', where ''x''''n'' ≠ ''x'' for all ''n'', we have ''f''(''x''''n'') → ''L''. Also, if ''f'' is a function on a first-countable space, then ''f'' is continuous if and only if whenever ''x''''n'' → ''x'', then ''f''(''x''''n'') → ''f''(''x'').

In first-countable spaces, sequential Compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which aren't compact (these are necessarily non-metric spaces). One such space is the Ordinal Space [0,ω1). 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