This post presents another example of a quotient space of a first countable space. The resulting quotient space is not first countable. After we first define the space without referring to the concepts of quotient space, we show that this example is a quotient space. This example will be further discussed when we discuss sequential spaces.
For a previous discussion on quotient space in this blog, see An example of a quotient space, I. For more information on quotient spaces in general, see .
Let be the set of all positive integers. For each and , let . Let . Let . For each , let . Define the space as:
The set is the horizontal part of the space and the set is the vertical part of the space. The origin is an additional point. In the topology for the space , each point in is isolated. Each point has an open neighborhood of the form
The open neighborhoods at are obtained by removing finitely many from and by removing finitely many isolated points in the that remain. The open neighborhoods just defined form a base for a topology on the set , i.e. by taking unions of these open neighborhoods, we obtain all the open sets for this space.
We wish to discuss and its subspace . If we consider as a topological space in its own right (open sets in are of the form where is open in ), then in there are no infinite compact sets (all compact subsets of are finite). In particular, no sequence of points can converge to the point . However, with respect to , is a limit point of . This implies that the space is not first countable.
A space is said to be a Frechet space if where , then there is some sequence that converges to the point . Any first countable space is a Frechet space. If there is a local base at and if , then whenever we pick , would converge to .
The space defined above is not a Frechet space. Note that and no sequence of points in can converge to .
Though the space defined above is not a first countable space, is a quotient space of a first countable space (in fact, a subspace of the Euclidean plane). Let where and are defined as in the definition of the space above. Let be the set . Let with the Euclidean topology inherited from the Euclidean plane.
Define a quotient space from by collapsing each pair of points into one point . This is done for each . For convenience, the point is shifted to . The resulting quotient space is the same as the space defined above.
The quotient space we just describe can also be described by the quotient map :
for each ,
for each ,
for each .
Then the following topology coincides with the topology on that we define earlier in the post:
- Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.