All spaces under consideration are Hausdorff. First countable spaces are those spaces where there is a countable local base at every point in the space. This is quite a strong property. For example, every first countable space that is also compact has a cap on its cardinality and the cap is the cardinality of the real line (the continuum). See The cardinality of compact first countable spaces, I in this blog. In fact, if the compact and first countable space is uncountable, it has cardinality continuum (see The cardinality of compact first countable spaces, III). Any metric space (or metrizable space) is first countable. In this post, we discuss the product of first countable spaces. In this regard, first countable spaces and metrizable spaces behave similarly. We show that the product of countably many first countable spaces is first countable while the product of uncountably many first countable is not first countable. For more information on the product topology, see .
The Product Space
Consider a collection of sets where . Let . The product is the set of all functions such that for each , . If the index set is finite, the functions can be regarded as sequences where each . If the index set , we can think of elements of the product as the sequence where each . In general we can regard as functions or as sequences .
Consider the topological spaces where . Let be the product as defined above. The product space of the spaces is with the topology defined in the following paragraph.
Let be the topology of each space , . Consider where for each , (i.e. is open in ) and for all but finitely many . The set of all such sets is a base for a topology on the product . This topology is called the product topology of the spaces , .
To more effectively work with product spaces, we consider a couple of equivalent bases that we can define for the product topology. Let be a base for the space . Consider such that there is a finite set where for each and for all . The set of all such sets is an equivalent base for the product topology.
Another equivalent base is defined using the projection maps. For each , consider the map such that for each in the product. In words, the function maps each point in the product space to its coordinate. The mapping is called the projection map. For each set , is the following set:
Consider sets of the form where is finite and is open in for each . The set of all such sets is another equivalent base for the product topology. If we only require that each , a predetermined base for the coordinate space , we also obtain an equivalent base for the product topology.
Countable Product of First Countable Spaces
For , let be a first countable space. We show that is a first countable space.
Let be the set of positive integers. For each , let and let be the set of all functions . For each and each , let be a countable local base at .
Let . We wish to define a countable local base at . For each , define to be:
Let be the set of all subsets of the product space of the following form:
where there is some such that and for all , .
Each is countable and is essentially the union of all the . Thus is countable. We claim that is a local base at . Let be an open set containing . We can assume that where there is some such that for each , is open in and for , .
For each , . Choose some such that . Let such that and for all . Then and . This completes the proof that is a first countable space.
Let be an uncountable index set. For , let . We want to avoid the situation that all but countably many are one-point space. So we assume each coordinate space has at least two points, say, and with . We show that is not first countable.
Let . Let be open subsets of the product space such that for each , . We show that there is some open set such that and each . For each , there is a basic open set such that .
Let . Since is uncountable and is countable, choose . Since has at least two points and , choose one of them that is different from , say, . Choose two disjoint open subsets and of such that and of . Let such that and for all . We have . For each , there is such that . Thus each . Thus there is no countable local base at . Thus any product space with uncountably many factors, each of which has at least two points, is never first countable.
- Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.