Topologies on the Cantor tree
by
Aleksander Błaszczyk
University of Silesia, Bankowa 14, 40-007 Katowice, Poland

We denote

Seq=w < w= È {nw: n < w}

and for s ∈ Seq and n ∈ w, s^\smallfrown n is the concatenation of s. For every A ⊆ w we set

s\smallfrown A={s\smallfrown n: n ∈ A}.

For every s ∈ Seq we fix a filter U_s of subsets of w. We assume that U_s contains the Frechét filter, i. e. 

{A ⊆ w: |w\A| < w} ⊆ Us.\tag*

(1)

For a collection \varSigma = {U_s: s ∈ Seq} of filters with property (*) the space S_\varSigma is the space with underlying set Seq and the topology defined by declaring W _ ( s W)( A _s)(s^ A W).

Equivalently: W _ ( s W)({n: s^ n W}_s).

Proposition 1

1. If U_s ={A ⊆ w: |w\A| < w} for each s ∈ Seq, then S_\varSigma is homeomorphic to the Arhangel'skii-Franklin space S_w.

2. For every collection \varSigma the space S_\varSigma has a continuous one-to-one mapping onto the space Q of all rationals numbers.

3. For every collection \varSigma the space S_\varSigma is zero-dimensional (countable) Hausdorff and dense in itself.

4. The space S_\varSigma is extremally disconnected iff the family \varSigma consists of ultrafilters.

For s ∈ Seq and x ∈ ∏{U_t: t ∈ Seq} we define a family {U(s, x, n): n <  w} as follows:

U(s, x, 0)={s},

U(s, x, n+1)= È {t\smallfrown x(t): t ∈ U(s, x, n)}.

Then the set

U(s, x)= È {U(s, x, n): n < w}

is an open subset of S_\varSigma. Indeed, if t ∈ U(s, x) then t ∈ U(s, x, n) for some n < w. Therefore x(t) ∈ U_t and t^\smallfrown x(t) ⊆ U(s, x, n+1) ⊆ U(s, x). In fact we have much more:

Lemma 1 The family

B\varSigma={U(s, x): s ∈ Seq and x ∈ Õ {Us: s ∈ Seq}}

consists of clopen sets and is a base of the space S_\varSigma .

For f, g ∈ ^ww we denote

f\preccurlyeq g⇔ (∃n < w)(∀k > n)(f(k) ≤ f(n))

and we recall the definition of the boundedness and dominating numbers: b={ - - : ^ ( g^)( f)(f g)} d={ - - : ^: ( f^)( g )(f g)} Obviously we have:

w1\leqslant b\leqslantd\leqslant 2w.

Theorem 1 For every collection \varSigma of filters and every s ∈ S_\varSigma we have

pc(s, S\varSigma)\geqslant b.

Since the set Seq is countable and dense in itself we have pc(x, S_\varSigma)=pw(x, S_\varSigma). Therefore we get

Corollary 1 For every collection \varSigma,

pw(S\varSigma)\geqslantb.

Theorem 2 If for every s ∈ Seq, U_s is a P-filter then

c(s, S\varSigma)\leqslant d+c(Us, w*).

One can easily construct in ZFC a filter U ∈ w^* such that U is a P-filter of the character ℵ₁. Hence we get

Corollary 2 If d=ℵ₁ then there exists a collection \varSigma such that c(s, S_\varSigma)=ℵ₁ for every s ∈ S_\varSigma.

Corollary 3 It is consistent with ZFC+¬CH that for some \varSigma, w(S_\varSigma)=ℵ₁.

If X is an infinite extremally disconnected compact space, then there is a homeomorphsm of X onto a nowhere dense subset of X. This fact is a consequence of the Balcar-Franek Theorem which says that every extremally disconnected compact space of weight k has a continuous mapping onto the Cantor cube of weight k.

Theorem 3 If the family \varSigma consists of pairwise non-comparable weak-P ultrafilters, then

(1) Every open surjection of bS_\varSigma onto itself is the identity.

(2) For every continuous injection f: bS_\varSigma→bS_\varSigma there exists a clopen set U ⊆ bS_\varSigma such that f\restriction U=id_U and f(bS_\varSigma\U) is nowhere dense.

Date received: May 25, 2008