The Cartesian product of the identity map and a quotient map can be a quotient map under one circumstance. We prove the following theorem.
Theorem 1
Let be a locally compact space. Let be a quotient map. Let the map be defined by for each . Then the map is a quotient map from to .
This is Theorem 3.3.17 in the Engelking topology text [1]. The theorem is attributed to J. H. C. Whitehead. The mapping defined in Theorem 1 is the Cartesian product of the identity map from to and the quotient map from onto . The theorem gives one circumstance in which the Cartesian product is also a quotient map. That is, taking the product of the identity map from a locally compact space to itself and a quotient map produces a quotient map. Potentially this gives us information about the product of the locally compact space in question and the space that is the quotient image. We give two natural applications of this theorem. Sequential spaces are precisely spaces that are quotient images of metric spaces (see here). The spaces called k-spaces are precisely the quotient images of locally compact spaces (see here). As corollary of Theorem 1, we show that the product of a locally compact metric space and a sequential space is a sequential space. In another corollary, we show that the product of a locally compact space and a k-space is a k-space. We have the following corollaries.
Corollary 2
Let be a locally compact metric space. Let be a sequential space. Then is a sequential space.
Corollary 3
Let be a compact metric space. Let be a sequential space. Then is a sequential space.
Corollary 4
Let be a locally compact space. Let be a k-space. Then is a k-space.
We give a proof of Theorem 1 and discuss the corollaries. We also give some examples.
Proof of Theorem 1
Let , and be the spaces described in the statement of Theorem 1, and let and be the mappings described in Theorem 1. To show that the map is a quotient map, we need to show that for any set , is an open set in if and only if is an open set in . Because the mapping is continuous, if is open in , we know that is an open set in . We only need to prove the other direction: if is an open set in , then is an open set in . To this end, let be an open set in and . We proceed to find some open set such that .
We choose and an open set with such that is compact and . We make the following observation,
- (1) for any , if and only if
Let . By observation (1), we have . Note that . Thus, . As a result, we have . We now need to show is an open subset of . Since is a quotient mapping, we know is open in if we can show is open in . The set is described as follows:
The last equality is due to Observation (1). Let be the projection map. Since is compact, the projection map is a closed map according to the Kuratowski Theorem (see here for its proof). Since is closed in , is closed in and is open in . It can be verified that . Thus, is open in . As a result, is open in . Furthermore, we have . This establishes that is open in . With that, the mapping is shown to be a quotient map.
Corollaries
Proof of Corollary 2
Let be a locally compact metric space and be a sequential space. According to the theorem shown here, is the quotient space of a metric space. There exists a metric space such that is the quotient image of . Let be a quotient map from onto . Consider the mapping defined by for all . By Theorem 1, is a quotient map. Since is the quotient image of the metric space , we establish that is a sequential space.
Corollary 3 follows from Corollary 2 since any compact space is a locally compact space.
Proof of Corollary 4
Let be a locally compact space and let be a k-space. According to the theorem shown here, there is a locally compact space such that is the quotient image of . Let be a quotient map from onto . Define by letting for all . According to Theorem 1, is a quotient map. Since is the quotient image of the locally compact space , we establish that is a k-space.
Sequential Fans
We illustrate the above corollaries using sequential fans. Sequential fans are sequential spaces. Products of sequential fans may no longer be sequential, in fact, may no longer be countably tight. In some cases, the tightness of a product of two sequential fans is dependent of your favorite set theory axiom (see here). However, the product of a sequential fan and a compact metric space is sequential.
Let be a convergent sequence including its limit. For convenience, denote such that each is isolated and an open neighborhood of consists of the point and all but finitely many . To make things more concrete, we can also let with the usual Euclidean topology. Let be an infinite cardinal number. Let be the topological sum of many copies of . The space is defined as with all the sequential limit points identified as one point called . The space is called the sequential fan with many spines. In , there are many copies of , which is called a spine.
Note that is a metric space. Because is the quotient image of , the sequential fan is a sequential space. In fact, is a Frechet space since it is a sequential space that does not contain a copy of the Arens’ space (see here). For the discussion of the Arens’ space, see here.
According to Corollary 3, the product of the sequential fan and a compact metric space is a sequential space. In particular, the product is always a sequential space. According to Corollary 4, is a k-space. The fact that the product is both a sequential space and a k-space is not surprising. Whenever the spaces and are sequential spaces, the product is a sequential space if and only if it is a k-space (see Theorem 2.2 [2]).
Reference
- Engelking R., General Topology, Revised and Completed edition, Elsevier Science Publishers B. V., Heldermann Verlag, Berlin, 1989.
- Tanaka Y., On Quasi-k-Spaces, Proc. Japan Acad., 46, 1074-1079, Berlin, 1970.
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