March | 2024 | Dan Ma's Topology Blog

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:

$B_f=\{ \infty \} \cup \{ (\alpha,n) \in \kappa \times \omega: n \ge f(\alpha) \}$

$f \in \omega^\omega$

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

Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
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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Daniel Ma Sequential Fan

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Daniel Ma Dominating Number

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Daniel Ma Bounding Number

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Daniel Ma quotient mapping

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