Monthly Archives: April 2023

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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Dan Ma quotient space
Daniel Ma quotient space

Dan Ma sequential space
Daniel Ma sequential space

Dan Ma k-space
Daniel Ma k-space

Dan Ma quotient mapping
Daniel Ma quotient mapping

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Daniel Ma topology

\copyright 2023 – Dan Ma

Defining Arens’ space using diagrams

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One way to define the Arens’ space is a 2-step approach, which is the quotient space approach. The first step is to identify an Euclidean plane consisting of convergent sequences (usually conveniently situated in the two-dimensional plane). The second step is to collapse certain points to make it a quotient space. Another way is to define the space directly, usually using an appropriate subset of the plane (of course, the resulting space is not an Euclidean space). We demonstrate both approaches using diagrams. In the first approach, we use two diagrams, the first one showing what the Euclidean space should look like, the second showing the resulting Arens’ space after certain points are identified. In the second approach, only one diagram is used (the standalone approach). The two-step approach is actually more informative since the quotient space of a separable metric space is a sequential space.

The following diagrams define the spaces without identifying specific points or locations in the Euclidean plane. The diagrams only indicate how the points relate to one another. For a definition of Arens’ space using the quotient space approach using specific points in the plane, see here. For a definition without connection to quotient space, see here. The red diagram and the blue diagram are for the quotient space approach (two-step). The pink diagram is the standalone approach.

The Arens’ space as discussed here is related to the Arens-Fort space, example 26 in Counterexamples in Topology [2].

The Red Diagram – The Euclidean Space

      \displaystyle \begin{aligned} & \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & S_5 & \text{ } & S_4 & \text{ } & S_3 & \text{ } & S_2 & \text{ } & S_1 \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } &\bullet & \text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet & \text{ } &\bullet & \text{ } & \bullet & \text{ } &\bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot &\text{ } &\cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot \\& \text{ } & \text{ } & \cdot &\text{ } & \cdot &\text{ } & \cdot & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & q_5 &\text{ } & q_4 & \text{ } & q_3 &\text{ } & q_2 & \text{ } & q_1 \\& \text{ } & \text{ } & \text{ } &\text{ } & \text{ } & \text{ } & \text{ } &\text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\& \bullet & \leftarrow & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet \\& p & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & p_5 &\text{ } & p_4 & \text{ } & p_3 &\text{ } & p_2 & \text{ } & p_1 \end{aligned}

In all 3 diagrams, the thick bullets represent points in the space and the tiny dots represent “dot dot dot”, indicating that the points or the sequences are continuing countably infinitely. The arrows in the diagrams point to the direction to which the points are converging.

The points in the red diagram form a subspace of the Euclidean plane. There are convergent sequences S_n going downward and going across from right to left. The point q_n is the limit of the sequence S_n. The sequence of points q_n converges to a point, which is ignored and not shown in the diagram. The points p_n are situated below the points q_n and converge to the point p. In this Euclidean space, the points in the sequences S_n are isolated points. An open set of the point q_n consists of q_n and all but finitely many points in the sequence S_n. Each point p_n is isolated. An open set of the point p consists of p and all but finitely many p_n.

The Blue Diagram – The Arens’ Space as a Quotient Space

      \displaystyle \begin{aligned} & \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & S_5 & \text{ } & S_4 & \text{ } & S_3 & \text{ } & S_2 & \text{ } & S_1 \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } &\bullet & \text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet & \text{ } &\bullet & \text{ } & \bullet & \text{ } &\bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot &\text{ } &\cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot \\& \text{ } & \text{ } & \cdot &\text{ } & \cdot &\text{ } & \cdot & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow \\& \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet \\& p & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & p_5 &\text{ } & p_4 & \text{ } & p_3 &\text{ } & p_2 & \text{ } & p_1 \end{aligned}

The blue diagram is established from the red diagram. The blue diagram is obtained by identifying the points q_n and p_n in the red diagram as one point called p_n. The resulting quotient space is the Arens’ space. Let S be the set of all points in the sequences S_n. Let P be the set of all points p_n. The Arens’ space is X=S \cup P \cup \{ p \} with the quotient topology. With the quotient topology, an open set containing the point p is obtained by removing finitely many vertical columns (a column is S_n plus the point p_n) from X and by removing finitely many points in the remaining columns. An open neighborhood of the point p_n consists of p_n and all but finitely many points in the sequence S_n. Points in the sequences S_n continue to be isolated points.

The Arens’ space is a sequential space since it is the quotient image of a separable metric space.

The Pink Diagram – The Arens’ Space as a Standalone Space

        \displaystyle \begin{aligned} & \\& \text{ } & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ }& \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & S_1 & \text{ } & S_2 & \text{ } & S_3 & \text{ } & S_4 & \text{ } & S_5 & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \text{ } &\text{ } & \text{ } & \text{ } & \text{ } &\text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\& \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& p & \text{ } & p_1 &\text{ } & p_2 & \text{ } & p_3 &\text{ } & p_4 & \text{ } & p_5 & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \end{aligned}

In the pink diagram, the bottom rows are the limit points. The point p_n is the sequential limit of the sequence S_n. The sequence S_n is displayed vertically. The convergence of the sequence S_n is not exhibited in the diagram and follows from how the open sets are defined.

Let S be the set of all points in the sequences S_n. Let P be the set of all points p_n. The Arens’ space is X=S \cup P \cup \{ p \} with the topology defined as follows. Each point in S is an isolated point. An open neighborhood of p_n \in P consists of the point p_n and all but finitely many points in S_n. An open neighbood containing the point p is obtained by removing finitely many vertical columns (a column is S_n plus the point p_n) from X and by removing finitely many points in the remaining columns.

Remarks

The blue diagram and the pink diagram are both representations of the Arens’ space. The space consists of countably many convergent sequences and their limits points plus one additional limit point called p. Both diagrams present the essential ideas without being tied to specific points or sequences in the Euclidean plane. Perhaps the diagrams will make it easier to think about the Arens’ space and remember the definition.

As mentioned earlier, the Arens’ space is a sequential space since it is the quotient space of a metric space (see the theorem here). Recall from above that the Arens’ space is X=S \cup P \cup \{ p \}. Clearly, p \in \overline{S}. Note that no sequence of points in S can converge to the point p. Thus, the Arens’ space is an example of a sequential space that is not a Frechet space. In fact, in order to know whether a sequential space W is Frechet, all we need to do is to determine if W contains a copy of the Arens’ space (see the theorem here). Thus, any space that is sequential but not Frechet contains a copy of the Arens’ space. In this case, Frechetness is characterized the absence of an Arens’ subspace. The Arens’ space is a canonical quotient space that appears in the characterization of other properties. See [1] for an example.

The property of being a sequential space is not hereditary. Consider the subspace of the Arens’ space Y=S \cup \{ p \}. As observed in the preceding paragraph, no sequence of points in S can converge to p. Thus, the set S is a sequentially closed set but not closed in Y. This means that Y is not a sequential space. Thus, the Arens’ space is a sequential that is not hereditarily sequential. In fact, a space is a Frechet space if and only if it is a hereditarily sequential space (see Theorem 1 here).

The subspace Y=S \cup \{ p \} discussed in the preceding paragraph is the Arens-Fort space, which is the example 26 in Steen and Seebach [2].

Reference

  1. Lin, S., A note on the Arens’ space and sequential fan, Topology Appl, 81, 185-196, 1997.
  2. Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.

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Dan Ma Arens’ space
Daniel Ma Arens’ space

Dan Ma sequential space
Daniel Ma sequential space

Dan Ma Frechet space
Daniel Ma Frechet space

Dan Ma quotient space
Daniel Ma quotient space

Dan Ma topology
Daniel Ma topology

\copyright 2023 – Dan Ma

Perfect preimages of Lindelof spaces

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Let f:X \rightarrow Y be a mapping from a topological space X onto a topological space Y. If f is a perfect map and Y is a Lindelof space, then so is X. If f is a closed map and Y is a paracompact space, then so is X. In other words, the pre-image of a Lindelof space under a perfect map is always a Lindelof space. Likewise, the pre-image of a paracompact space under a closed map is always a paracompact space. After proving these two facts, we show that for any compact space Y, the product X \times Y is Lindelof (paracompact) for any Lindelof (paracompact) space X. All spaces under consideration are Hausdorff.

Another way to state the above two facts is that Lindelofness is an inverse invariance under perfect maps and that paracompactness is an inverse invariance of the closed maps. In general, a topological property is an inverse invariance of a class of mappings \mathcal{M} if the following holds: for any mapping f:X \rightarrow Y belonging to \mathcal{M}, if Y has the property, then so does X. In contrast, a topological property is an invariance of a class of mappings \mathcal{M} if for any mapping f:X \rightarrow Y belonging to \mathcal{M}, if X has the property, then so does Y.

All mappings under consideration are continuous maps. A mapping f:X \rightarrow Y, where f(X)=Y, is a closed map if for any closed subset A of X, f(A) is closed in Y. A mapping f:X \rightarrow Y, where f(X)=Y, is a perfect map if f is a closed map and that the point inverse f^{-1}(y) is compact for each y \in Y.

Perfect mappings and closed mappings are objects with strong properties. Such a map places a restriction on what topological properties the “domain” space or the “range” space can have. The theorems below indicate that it is not possible to map a non-Lindelof space onto a Lindelof space using a perfect map and that it is not possible to map a non-paracompact space onto a paracompact space using a closed map. On the other hand, it is not possible to map a separable metric space onto a separable but non-metric space using a perfect map (see here). We prove the following theorems.

Theorem 1…. Lindelofness (or the Lindelof property) is an inverse invariant of the perfect maps.

Theorem 2 ….Paracompactness is an inverse invariant of the closed maps.

Lemma 3 …. Let f:X \longrightarrow Y be a closed map with f(X)=Y. Let V be an open subset of X. Define f_*(V)=\{ y \in Y: f^{-1}(y) \subset V \}. Then the set f_*(V) is open in Y and that f_*(V) \subset f(V).

For the proof of Lemma 3, see Lemma 2 here.

Proof of Theorem 1
Let f:X \longrightarrow Y be a perfect map with f(X)=Y. Suppose Y is Lindelof. Let \mathcal{U} be an open cover of X. Without loss of generality, we can assume that \mathcal{U} is closed under finite unions. For each U \in \mathcal{U}, define f_*(U)=\{ y \in Y: f^{-1}(y) \subset U \}. By Lemma 3, each f_*(U) is an open subset of Y. We claim that \mathcal{V}=\{ f_*(U): U \in \mathcal{U} \} is an open cover of Y. To this end, let y \in Y. Since f is a perfect map, the point inverse f^{-1}(y) is compact. As a result, we can find a finite \mathcal{F} \subset \mathcal{U} such that f^{-1}(y) \subset \bigcup \mathcal{F}=W. Since \mathcal{U} is closed under finite unions, W \in \mathcal{U}. It follows that y \in f_*(W). Since Y is Lindelof, there exists a countable \{W_0,W_1,W_2,\cdots \} \subset \mathcal{V} such that Y = \bigcup_{n=0}^\infty W_n. For each n, W_n=f_*(U_n) for some U_n \in \mathcal{U}. We claim that \{U_0,U_1,U_2,\cdots \} is a cover of X. To this end, let x \in X. Then for some n, y=f(x) \in W_n=f_*(U_n). This implies that x \in f^{-1}(y) \subset U_n. Thus, the open cover \mathcal{U} has a countable subcover. This concludes the proof of Theorem 1. \square

Proof of Theorem 2
Let f:X \longrightarrow Y be a perfect map with f(X)=Y. Suppose Y is paracompact. Let \mathcal{U} be an open cover of X. For each U \in \mathcal{U}, define f_*(U) as in Lemma 3. By Lemma 3, each f_*(U) is an open subset of Y. Let \mathcal{V}=\{ f_*(U): U \in \mathcal{U} \}. As shown in the proof of Theorem 1, \mathcal{V} is an open cover of Y. Since Y is paracompact, there exists a locally finite open refinement \mathcal{W} of \mathcal{V}. Let \mathcal{W}_0=\{ f^{-1}(W): W \in \mathcal{W} \}.

We show three facts about \mathcal{W}_0. (1) It is an open cover of X. (2) It is a locally finite collection in X. (3) It is a refinement of \mathcal{U}. To see (1), note that \mathcal{W} is an open cover of Y. As a result, \mathcal{W}_0 is an open cover of X. To see (2), let x \in X. We find an open O \subset X such that x \in O and such that O intersects only finitely many elements of \mathcal{W}_0. Since \mathcal{W} is locally finite in Y, there exists an open B \subset Y such that y=f(x) \in B and such that B intersects only finitely many elements of \mathcal{W}, say, W_0,W_1,\cdots,W_n. Let O=f^{-1}(B). Clearly, x \in O. It can be verified that the only elements of \mathcal{W}_0 having non-empty intersections with O are f^{-1}(W_0),f^{-1}(W_1),\cdots,f^{-1}(W_n). To see (3), let f^{-1}(W) \in \mathcal{W}_0 where W \in \mathcal{W}. Then W \subset V=f_*(U) for some V \in \mathcal{V} and some U \in \mathcal{U}. We claim that f^{-1}(W) \subset U. Let x \in f^{-1}(W). Then y=f(x) \in W \subset V=f_*(U). This implies that x \in f^{-1}(y) \subset U. It follows that \mathcal{W}_0 is a locally finite open refinement of the open cover \mathcal{U}. This completes the proof of Theorem 2. \square

Productively Paracompact Spaces

A space X is productively paramcompact if X \times Y is paracompact for every paracompact space Y. The definition for productively Lindelof can be stated in a similar way. For some reason, the term “productively paracompact” is not used in the literature but is a topic that had been extensively studied. It is also a topic found in this site. The following four classes of spaces are productively paracompact (see here and here).

  • Compact spaces
  • \sigma-compact spaces
  • Locally compact spaces
  • \sigma-locally compact spaces

The proof for compact spaces being productively paracompact given here uses the Tube Lemma (see here). As applications of Theorem 1 and Theorem 2, we use the two theorems to show that compact spaces are both productively Lindelof and productive paracompact.

Theorem 4…. Let Y any compact space. Then X \times Y is Lindelof for every Lindelof space X.

Theorem 5 ….Let Y any compact space. Then X \times Y is paracompact for every paracompact space X.

Theorems 4 and 5 are corollaries to the Kuratowski theorem (see here) and Theorems 1 and 2 above. Suppose Y is compact. Then the projection map from X \times Y onto X is a closed map. The paracompactness of X \times Y follows whenever X is paracompact. The projection map is also perfect since the point inverses are compact due to the compactness of the factor Y. Then the Lindelofness of X \times Y follows whenever X is Lindelof.

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Daniel Ma perfect map

Dan Ma paracompact space
Daniel Ma paracompact space

Dan Ma Lindelof space
Daniel Ma Lindelof space

Dan Ma topology
Daniel Ma topology

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