It is well known that in a compact Hausdorff space, if every point in the space is a point, then the space is first countable (see The cardinality of compact first countable spaces, II). Outside of compact spaces, this is not true. We present an example of a space where every point is a point but has no countable local base at any one point. This is an opportunity to introduce the notion of quotient space. Our example can be viewed as the image of a first countable space under a quotient map. We first define the example and then present a brief discussion of quotient spaces.
Let be the set of all positive integers. Let . Let and . Consider as a subspace of the Eucliean plane. In the Euclidean topology, each point is is isolated and an open neighborhood at each point in is of the form:
We use to define a space by considering as one point (call this point ). Thus . Points in remain isolated. An open neighborhood of is of the form where and is a Euclidean open subset of such that .
To facilitate the discussion, we describe the open sets at the point in more details. Let be the set of all functions . For each and , let . An open neighborhood of is of the following form:
To describe in words, each open neighborhood at is obtained by removing finitely many points in each vertical segment .
The resulting space is not first countable. We show that any countable set of open neighborhoods at cannot be a local base at . To this end, we consider where each . We wish to find an open set such that and each is not a subset of .
For each , pick some such that . Consider . Note that for each , and . This shows that no countable number of open neighborhoods can form a base at the point .
However, every point in the space is a point. For the point , we have where is a constant function mapping to .
We now present one definition of quotient spaces. Let be a topological space. Let be a surjection, i.e. . Consider the following collection of subsets of :
The set is a topology for the set . With this topology on , the function is continuous. In fact, is the finest (largest or strongest) topology on that makes the function continuous, i.e. if is another topology on making a continuous function, then .
The topology is called the quotient topology induced on by the mapping . When is given such a quotient topology, it is called a quotient space of (most of the times just called quotient space). The induced map is called a quotient map. If has a quotient topology defined by a quotient map, then is said to be the quotient image of . We plan to discuss quotient space and quotient topology in greater details in this blog. For further information, see .
In the example discussed in this post, the quotient map is to map each point in to itself and to map each point in to a single point . In essence, we collapse the whole x-axis into one single point. The open sets for the point are simply the Euclidean open sets containing . This example shows that the quotient image of a first countable space needs not be first countable. Though the quotient image of a first countable space may not be first countable, it has the property that sequences suffice to define the topology (these are called sequential spaces, see ).
- Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.