Category Archives: Mappings

The product of the identity map and a quotient map

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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 X be a locally compact space. Let q:Y \rightarrow Z be a quotient map. Let the map f:X \times Y \rightarrow X \times Z be defined by f(x,y)=(x,q(y)) for each (x,y) \in X \times Y. Then the map f is a quotient map from X \times Y to X \times Z.

This is Theorem 3.3.17 in the Engelking topology text [1]. The theorem is attributed to J. H. C. Whitehead. The mapping f defined in Theorem 1 is the Cartesian product of the identity map from X to X and the quotient map from Y onto Z. 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 X be a locally compact metric space. Let Y be a sequential space. Then X \times Y is a sequential space.

Corollary 3
Let X be a compact metric space. Let Y be a sequential space. Then X \times Y is a sequential space.

Corollary 4
Let X be a locally compact space. Let Y be a k-space. Then X \times Y 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 X, Y and Z be the spaces described in the statement of Theorem 1, and let q and f be the mappings described in Theorem 1. To show that the map f is a quotient map, we need to show that for any set O \subset X \times Z, O is an open set in X \times Z if and only if f^{-1}(O) is an open set in X \times Y. Because the mapping f is continuous, if O is open in X \times Z, we know that f^{-1}(O) is an open set in X \times Y. We only need to prove the other direction: if f^{-1}(O) is an open set in X \times Y, then O is an open set in X \times Z. To this end, let f^{-1}(O) be an open set in X \times Y and (a,b) \in O. We proceed to find some open set U \times V \subset X \times Z such that (a,b) \in U \times V \subset O.

We choose c \in q^{-1}(b) and an open set U \subset X with a \in U such that \overline{U} is compact and \overline{U} \times \{ c \} \subset f^{-1}(O). We make the following observation,

  • (1) for any y \in Y, \overline{U} \times \{ y \} \subset f^{-1}(O) if and only if \overline{U} \times q^{-1} q(y) \subset f^{-1}(O)

Let V=\{ z \in Z: \overline{U} \times q^{-1}(z) \subset f^{-1}(O) \}. By observation (1), we have \overline{U} \times q^{-1}q(c) \subset f^{-1}(O). Note that q(c)=b. Thus, b \in V. As a result, we have (a,b) \in U \times V \subset O. We now need to show V is an open subset of Z. Since q is a quotient mapping, we know V is open in Z if we can show q^{-1}(V) is open in Y. The set q^{-1}(V) is described as follows:

    \displaystyle \begin{aligned} q^{-1}(V)&=\{ y \in Y: q(y) \in V \} \\&=\{y \in Y: \overline{U} \times q^{-1}q(y) \subset f^{-1}(O) \} \\&=\{y \in Y: \overline{U} \times \{ y \} \subset f^{-1}(O) \} \end{aligned}

The last equality is due to Observation (1). Let \pi: \overline{U} \times Y \rightarrow Y be the projection map. Since \overline{U} is compact, the projection map \pi is a closed map according to the Kuratowski Theorem (see here for its proof). Since (\overline{U} \times Y) \backslash f^{-1}(O) is closed in \overline{U} \times Y, C=\pi(\overline{U} \times Y \backslash f^{-1}(O)) is closed in Y and Y \backslash C is open in Y. It can be verified that q^{-1}(V)=Y \backslash C. Thus, q^{-1}(V) is open in Y. As a result, V is open in Y. Furthermore, we have (a,b) \in U \times V \subset O. This establishes that O is open in X \times Z. With that, the mapping f is shown to be a quotient map. \square

Corollaries

Proof of Corollary 2
Let X be a locally compact metric space and Y be a sequential space. According to the theorem shown here, Y is the quotient space of a metric space. There exists a metric space M such that Y is the quotient image of M. Let q:M \rightarrow Y be a quotient map from M onto Y. Consider the mapping f:X \times M \rightarrow X \times Y defined by f(x,y)=(x,q(y)) for all (x,y) \in X \times M. By Theorem 1, f is a quotient map. Since X \times Y is the quotient image of the metric space X \times M, we establish that X \times Y is a sequential space. \square

Corollary 3 follows from Corollary 2 since any compact space is a locally compact space.

Proof of Corollary 4
Let X be a locally compact space and let Y be a k-space. According to the theorem shown here, there is a locally compact space W such that Y is the quotient image of W. Let q:W \rightarrow Y be a quotient map from W onto Y. Define f:X \times W \rightarrow X \times Y by letting f(x,y)=(x,q(y)) for all (x,y) \in X \times W. According to Theorem 1, f is a quotient map. Since X \times Y is the quotient image of the locally compact space X \times W, we establish that X \times Y is a k-space. \square

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 S be a convergent sequence including its limit. For convenience, denote S=\{ q_0,q_1,q_2,\cdots \} \cup \{ q \} such that each q_n is isolated and an open neighborhood of q consists of the point q and all but finitely many q_n. To make things more concrete, we can also let S=\{ 1,\frac{1}{2},\frac{1}{3},\cdots \} \cup \{ 0 \} with the usual Euclidean topology. Let \kappa be an infinite cardinal number. Let M(\kappa) be the topological sum of \kappa many copies of S. The space S(\kappa) is defined as M(\kappa) with all the sequential limit points identified as one point called \infty. The space S(\kappa) is called the sequential fan with \kappa many spines. In S(\kappa), there are \kappa many copies of S \cup \{ \infty \}, which is called a spine.

Note that M(\kappa) is a metric space. Because S(\kappa) is the quotient image of M(\kappa), the sequential fan S(\kappa) is a sequential space. In fact, S(\kappa) 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 S(\kappa) and a compact metric space is a sequential space. In particular, the product S(\kappa) \times S is always a sequential space. According to Corollary 4, S(\kappa) \times S is a k-space. The fact that the product is both a sequential space and a k-space is not surprising. Whenever the spaces X and Y are sequential spaces, the product X \times Y is a sequential space if and only if it is a k-space (see Theorem 2.2 [2]).

Reference

  1. Engelking R., General Topology, Revised and Completed edition, Elsevier Science Publishers B. V., Heldermann Verlag, Berlin, 1989.
  2. Tanaka Y., On Quasi-k-Spaces, Proc. Japan Acad., 46, 1074-1079, Berlin, 1970.

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