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Sequential fan and the dominating number

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The sequential fan S(\kappa) is the quotient space obtained by identifying the limit points of a topological sum of \kappa many convergent sequences. We focus on S(\omega), the sequential fan derived from countably and infinitely many convergent sequences. Because only countably many convergent sequences are used, S(\omega) is intimately connected to the combinatorics in \omega^\omega, which is the family of all functions from \omega into \omega. In particular, we show that the character at the limit point \infty in S(\omega) equals to the dominating number \mathfrak{d}. The dominating number \mathfrak{d} and the bounding number \mathfrak{b}, introduced below, are quite sensitive to set theoretic assumptions. As a result, pinpointing the precise cardinality of the character of the point \infty in the sequential fan S(\omega) requires set theory beyond ZFC. The fact that the character at \infty in the sequential fan S(\omega) is identical to the dominating number \mathfrak{d} is mentioned in page 13 in chapter a-3 of the Encyclopedia of General Topology [1].

Sequential fans had been discussed previously (see here). See here, here, here, here, and here for previous discussion on the bounding number and the dominating number.

Sequential Fans

As mentioned above, a sequential fan is the quotient space on a disjoint union of convergent sequences with all the limit points of the sequences collapsed to one point called \infty. We first give a working definition. To further provide intuition, we also show that our sequential fan of interest S(\omega) is the quotient space of a subspace of the Euclidean plane (i.e., the countably many convergent sequences can be situated in the plane).

In the discussion that follows, \omega is the set of all non-negative integers. The set \omega^\omega is the family of all functions from \omega into \omega. Let \kappa be an infinite cardinal number. The sequential fan S(\kappa) with \kappa many spines is the set S(\kappa)=\{ \infty \} \cup (\kappa \times \omega) with the topology defined as follows:

  • Every point in \kappa \times \omega is made an isolated point.
  • An open neighborhood of the point \infty is of the following form:
      • B_f=\{ \infty \} \cup \{ (\alpha,n) \in \kappa \times \omega: n \ge f(\alpha) \} where f \in \omega^\omega

In this formulation of the sequential fan, the set \{(\alpha, n): n \in \omega \}, where \alpha < \kappa, is a sequence converging to \infty. For each such convergent sequence, the open neighborhood B_f contains all but finitely many points.

Our focus is S(\omega), where S(\omega)=\{ \infty \} \cup (\omega \times \omega).

A View From the Euclidean Plane

The formulation of the sequential fan S(\kappa) given above is a good working formulation. We now describe how S(\omega) can be derived from the Euclidean plane. Consider the following diagram.

        \displaystyle \begin{aligned} & \\& \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{ } & \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{ } & \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{ } & \downarrow &\text{ } & \downarrow & \text{ } & \downarrow &\text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } \\& \text{ } & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \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 above diagram, the thick bullets are the 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. There are countably and infinitely many convergent sequences, named S_1,S_2,S_3,\cdots, with p_n being the limit of the sequence S_n. For convenience, we can let p_n be the point (n,0) in the plane and S_n be a sequence converging downward to p_n. Let S=S_1 \cup S_2 \cup S_3 \cup \cdots and let P=\{p_1,p_2,p_3,\cdots \}. Consider the space X=S \cup P with the topology inherited from the Euclidean plane. Any point in any one of the convergent sequence S_n is an isolated point. An open neighborhood of the limit point p_n consists of p_n and all but finitely many points in S_n.

The diagram and the preceding paragraph set up the scene. We are now ready to collapse points (or define the quotient map). We collapse the set of all limit points P to one point called \infty. The resulting quotient space is Y=S \cup \{ \infty \}. In this quotient space, S is the set of all points in the countably many convergent sequences with each point isolated. An open neighborhood at \infty consists of \infty and all but finitely many points in each convergent sequence. This formulation is clearly equivalent to the sequential fan S(\omega) formulated earlier.

When \kappa is uncountable, the topological sum of \kappa many convergent sequences can no longer viewed in a Euclidean space. However, the topological sum is still a metric space (just not a separable one). We can still collapse the limit points into one point called \infty. The resulting quotient space is identical to S(\kappa) formulated above.

The Combinatorics on the Integers

We begin the combinatorics by defining the order \le^*. Recall that \omega^\omega is the family of all functions from \omega into \omega. For f,g \in \omega^\omega, declare f \le^* g if f(n) \le g(n) for all but finitely many n \in \omega. We write f \not \le^* g if the negation of f \le^* g is true, i.e., f(n)>g(n) for infinitely many n \in \omega. The order \le^* is a reflexive and transitive relation.

A set F \subset \omega^\omega is said to be bounded if F has an upper bound according to the order \le^*, i.e., there exists g \in \omega^\omega such that f \le^* g for all f \in F (Here, g is the upper bound of F). The set F is said to be unbounded if it is not bounded according to \le^*. That is, F is unbounded if for each g \in \omega^\omega, there exists f \in F such that f \not \le^* g. A set F \subset \omega^\omega is said to be a dominating set if F is cofinal in \le^*, i.e., for each f \in \omega^\omega, there exists g \in F such that f \le^* g. We now define two cardinal numbers as follows:

    \mathfrak{b}=\text{min} \{ \lvert F \lvert: F \subset \omega^\omega \text{ is unbounded} \}

    \mathfrak{d}=\text{min} \{ \lvert F \lvert: F \subset \omega^\omega \text{ is dominating} \}

The first number \mathfrak{b} is called the bounding number and the second one \mathfrak{d} is called the dominating number. Both are upper bounded by the continuum \mathfrak{c}, i.e., \mathfrak{b} \le \mathfrak{c} and \mathfrak{d} \le \mathfrak{c}. Using a diagonal argument, we can show that both of these cardinal numbers are not countable. Thus, we have \omega_1 \le \mathfrak{b},\mathfrak{d} \le \mathfrak{c}. How do \mathfrak{b} and \mathfrak{d} relate? We have \mathfrak{b} \le \mathfrak{d} since every dominating set is also an unbounded set.

The Character at Infinity

The sequential fan S(\omega) is not first countable at the point \infty. In other word, there does not exist a countable local base at \infty. To see this, let \{ B_{f_1},B_{f_2},B_{f_3},\cdots \} be a countable collection of open neighborhoods of \infty. Using a diagonal argument, we can find f \in \omega^\omega such that B_{f_n} \not \subset B_f for all n. This shows that no countable collection of open neighborhoods can be a base at \infty. Thus, the character at \infty must be uncountable (the character at a point is the minimum cardinality of a local base at the point). Thus, we have have \chi(S(\omega),\infty)>\omega. Furthermore, we have \omega_1 \le \chi(S(\omega),\infty) \le \mathfrak{c} (character is at least \omega_1 but no more than continuum). The range from \omega_1 to continuum \mathfrak{c} is a narrow range if continuum hypothesis holds, but can be a large range if continuum hypothesis does not hold. Can we pinpoint the character at \infty more narrowly and more precisely?

Connecting the Dominating Number to the Sequential Fan

We claim the for the sequential fan S(\omega), the character at the point \infty is the dominating number \mathfrak{d} introduced above. To establish this claim, we set up a different formulation of dominating set. A set F \subset \omega^\omega is said to be a special dominating set if for each f \in \omega^\omega, there exists g \in F such that f(n) \le g(n) for all n \in \omega. We define the cardinal number \mathfrak{d}_1 as follows:

    \mathfrak{d}_1=\text{min} \{ \lvert F \lvert: F \subset \omega^\omega \text{ is a special dominating set} \}

Note that the term “special dominating” is not a standard term. It is simply a definition that facilitates the argument at hand. One key observation is that when F is a special dominating set, the collection \{B_f: f \in F \} becomes a base at the point \infty. Since the cardinal number \mathfrak{d}_1 is the minimum cardinality of a base at \infty, we only need to show that \mathfrak{d}=\mathfrak{d}_1. Since every special dominating set is a dominating set, we have \mathfrak{d} \le the cardinality of every special dominating set. Thus, \mathfrak{d} \le \mathfrak{d}_1.

Next we show \mathfrak{d}_1 \le \mathfrak{d}. To this end, we show that \mathfrak{d}_1 \le the cardinality of every dominating set. We claim that for every dominating set F, there exists a special dominating set F_* with \lvert F_* \lvert=\lvert F \lvert. Once this is established, we have \mathfrak{d}_1 \le the cardinality of every dominating set and thus \mathfrak{d}_1 \le \mathfrak{d}.

Let F be a dominating set. For each n \in \omega with n \ge 1, define the following:

  • D_n=\{0,1,\cdots,n-1 \}
  • E_n=\{ n,n+1,n+2,\cdots \}
  • A_n=\omega^{D_n}
  • B_n=\omega^{E_n}

If h \in A_n and k \in B_n, then we take h \cup k to be a function in \omega^\omega. For each n \ge 1 and for each f \in F, define the following:

    F_{f,n}=\{h \cup (f \upharpoonright E_n): h \in A_n \}

with f \upharpoonright E_n representing the function f restricted to the set E_n. Let F_*=\bigcup \{F_{f,n}: n \ge 1, f \in F \}. Note that each F_{f,n} is countable. As a result, \lvert F_* \lvert=\lvert F \lvert. Because F is a dominating set, F_* is a special dominating set. We have just established that \mathfrak{d}_1 = \mathfrak{d} and that the character of the point \infty in the sequential fan S(\omega) is the dominating number \mathfrak{d}.

Remarks

Can we pinpoint the character at \infty? The answer is a partial yes. We establish that \chi(S(\omega),\infty)=\mathfrak{d}. However, the dominating number and the bounding number as well as other small cardinals are very sensitive to set theory. For example, when continuum hypothesis (CH) holds, The dominating number \mathfrak{d} is continuum. Thus, it is consistent with ZFC that \chi(S(\omega),\infty) is continuum. It is also consistent with ZFC that \omega_1 \le \mathfrak{b} <\mathfrak{d}<\mathfrak{c}. Thus it is consistent that \chi(S(\omega),\infty) is greater than \omega_1 and less than continuum. Though the dominating number tells us how big the character at \infty is, we cannot pinpoint precisely where the character is in the range between \omega_1 and continuum. For more information about dominating number and other small cardinals, see chapter 3 in the Handbook of Set-Theoretic Topology [2].

The fact that the character at \infty in the sequential fan S(\omega) is identical to the dominating number \mathfrak{d} is mentioned in page 13 in chapter a-3 of the Encyclopedia of General Topology [1].

The sequential fan S(\omega) is a space that has a simple definition. After all, the starting point is a subspace of the Euclidean plane with S(\omega) obtained by collapsing the limit points. Though the space is very accessible, the size of the character at the limit point \infty is unknowable if we work only in ZFC. It is a short “distance” from the definition of the sequential fan S(\omega) to the set-theoretic unknowable statement. This makes the sequential fan S(\omega) an interesting example and an excellent entry point of learning more set-theoretic topology.

Reference

  1. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
  2. Van Douwen, E. K., The Integers and Topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, (1984) 111-167.

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    Dan Ma Sequential Fan
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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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