Predavanje za konferenciju

THE II WORKSHOP ON COVERINGS, SELECTIONS AND GAMES IN TOPOLOGY 19.12-22.12, 2005 Lecce V. Pavlovi´ c

Games and a property of Pytkeev University of Niˇs, Serbia and Montenegro

Games and a property of Pytkeev

2

A space X is said to have the Reznichenko property at the point x ∈ X (see [4],[6]) provided that x ∈ A\A and A ⊆ X imply the existence of a sequence (An : n ∈ N) of pairwise disjoint finite subsets of A such that for every neighborhood V of x there is a n0 ∈ N with ∀n ≥ n0 (V ∩ An 6= ∅). If X has that property at all of its points then X is simply said to have the Reznichenko property.

The Reznichenko property in function spaces Ck (X) and Cp (X) was studied e.g. in [9], [4] and [8] and in the context of hyperspaces in [3].

Vladimir Pavlovi´ c

December 14, 2005


3

We say that X has the Pytkeev property at the point a ∈ X (see e.g. [6]) if a ∈ A \ A implies the existence of a π-network at a consisting of countably infinite subsets of A. If X has the Pytkeev property at each of its points, X is simply said to have the Pytkeev property.


December 14, 2005


4

Definition 1 A space X is said to have the selectively Reznichenko property at the point x ∈ X if for each sequence (An : n ∈ N) of subsets of X with x ∈ An \ An for all n ∈ N, there is a sequence (Bn : n ∈ N) of pairwise disjoint sets such that each Bn is a finite subset of An and such that for each open neighborhood V of x there is an n0 ∈ N with ∀n ≥ n0 (V ∩ Bn 6= ∅). If X has that property at all of its points then X is simply said to have the selectively Reznichenko property.


December 14, 2005


5

Definition 2 For two sets A and F (we look at F as a list of properties) the formula R(A, F ) abbreviates the statement: for each sequence (An : n ∈ N) of elements of A there is a sequence (Bn : n ∈ N), where each Bn is a finite subset of An , such that the Bn -s are pairwise disjoint and ∀F ∈ F ∃n ∈ N ∀m ≥ n∃S ∈ Bm (S ∈ F ).


December 14, 2005


6

If a ∈ X denote by Ωa = Ωa (X) the family of all A ⊆ X such that a ∈ A \ A, and fix a local base Ba at a. Then R(Ωa , Ba ) is another way of saying that X has the selectively Reznichenko property at a.


December 14, 2005


7

A space X has the selectively Pytkeev property at the point a ∈ X if for each sequence (An : n ∈ N) of elements of Ωa , there is a sequence (Bn : n ∈ N), where each Bn is an countably infinite subset of An , such that {Bn : n ∈ N} is a π-network at a. If X has the selectively Pytkeev property at each of its points, X is simply said to have the selectively Pytkeev property.


December 14, 2005


8

As it is trivial to see, the selectively Pytkeev property at a ∈ X can be given by the following equivalent formulation: for each sequence (An : n ∈ N) of elements of Ωa , there is a sequence (Bn : n ∈ N), where each Bn is a countably infinite subset of An , such that for each Q f ∈ (Bn : n ∈ N) the set {f (n) : n ∈ N} is an element of Ωa . We now introduce a new selection hypothesis generalizing the selectively Pytkeev property.


December 14, 2005


9

Definition 3 For two sets A and B the formula P (A, B) abbreviates the statement: for each sequence (An : n ∈ N) of elements of A there is a sequence (An : n ∈ N), where each An is a countably infinite subset of An , such Q that for each f ∈ (An : n ∈ N) the set {f (n) : n ∈ N} is an element of B. It is clear that X has the selectively Pytkeev property at a point a ∈ X if and only if P (Ωa , Ωa ) holds.


December 14, 2005


10

The infinitely long game GameP (A, B) is defined as follows: Two players, W(hite) and B(lack), play a round for each natural number n. In the first round player W plays A1 ∈ A and B then responds by a1 = (a11 ), where a11 ∈ A1 . In the n-th round W plays An ∈ A and B n n , . . . , a ), where a then responds by an = (an 1 n i ∈ Ai for all 1 ≤ i ≤ n. A play (An , an : n ∈ N) is won by B if the sequence (an m : n ≥ m) is injective for all m ∈ N, and if for each f : N → N with f (n) ≥ n, the set {afn(n) : n ∈ N} is an element of B. Otherwise it is won by W.


December 14, 2005


11

We shall write GameP yt(a) to shorten GameP (Ωa , Ωa ), or simply GameP yt whenever the point a at which the game is played is understood.


December 14, 2005


12

A space X is said to be strictly Fr´ echet at the point x ∈ X (see e.g. [9]) if for each sequence (An : n ∈ N) of subsets of X with x ∈ An \ An for all n ∈ N, there is a sequence (an : n ∈ N), with an ∈ An , converging to x. If X has that property at all of its points then X is simply said to be strictly Fr´ echet.


December 14, 2005


13

Definition 4 For two sets A and F (the latter should be regarded as a list of certain properties) the statement C(A, F ) means that for each sequence (Un : n ∈ N) of elements of A there is a sequence (Vn : n ∈ N) such that each Vn ∈ Un and ∀F ∈ F ∃n0 ∀n ≥ n0 (Vn ∈ F ). Obviously X is strictly Fr´ echet at a ∈ X if and only if C(Ωa , Ba ) holds, where Ωa and Ba are as previously mentioned.


December 14, 2005


14

Proposition 1 Let X be a T1 space. If X is strictly Fréchet then it has the selectively Pytkeev property.

Proposition 2 If a T1 space X is strictly Fréchet then it has the selectively Reznichenko property.


December 14, 2005


15

Proposition 3 If Ck (X) (Cp (X)) has the selectively Pytkeev property then it has the selectively Reznichenko property.


December 14, 2005


16

Definition 5 ([9]) An open cover U is called ω-shrinkable if there is a function f such that for each U ∈ U , f (U ) is a cozero set, f (U ) ⊆ U and {f (U ) : U ∈ U } is an ω-cover.


December 14, 2005


17

Definition 6 A family U of subsets of a space X is said to be an “3-kshrinkable cover” of X if there is a function g such that for each U ∈ U g(U ) = (VU , ZU ), where VU ⊆ ZU ⊆ U , VU is a cozero set, ZU is a zero set and {VU : U ∈ U } k- covers X; it is called an open ”3-k-shrinkable cover” provided that its elements are open subsets of X. The collection of all non trivial open 3-k-shrinkable covers of X will be denoted by 3-Kshr (X) or just by 3-Kshr when it is clear to which X the notation refers.


December 14, 2005


18

Theorem 1 Ck (X) has the selectively Reznichenko property if and only if R(3 − Kshr , C) holds.

Theorem 2 Ck (X) is strictly Fréchet if and only if C(3 − Kshr , C) holds.


December 14, 2005


19

Theorem 3 For a space X, the implications (i) ⇒ (i + 1), i = 1, 4 between the statements (1) − (5) listed bellow are true: (1) Ck (X) is strictly Fréchet; (2) ∗ GameP (K(X), K(X)); (3) ∗ GameP (3 − Kshr (X), K(X)); (4) the player W has no winning strategy in the game GameP yt(a) played on Ck (X) at any a ∈ Ck (X); (5) Ck (X) has the selectively Pytkeev property.


December 14, 2005


20

Theorem 4 The statements of Theorem 3 are equivalent for a locally compact X.


December 14, 2005


21

Theorem 5 For a locally compact space X the following are equivalent: (1) Ck (X) is strictly Fréchet; (2) Ck (X) has the selectively Pytkeev property; (3) Ck (X) has the selectively Reznichenko property.


December 14, 2005


22

Lemma 1 Let X be such a space that each open k-cover has a countable k-subcover. If > 0 and o ∈ A ⊆ Ck (X) then there is a B ⊆ A and a function s : B → (0, ) such that at least one of the families {|f |← [0, s(f )) : f ∈ B} and {|f |← [0, s(f )] : f ∈ B} is a 3-k- shrinkable open cover of X.


December 14, 2005


23

Corollary 1 If X is locally compact and Ck (X) has the selectively Reznichenko property, then X must be σ-compact.


December 14, 2005


24

References [1] R. Engelking, General Topology, PWN, Warszawa, 1977. [2] Lj.D.R. Koˇ cinac Closure properties of function spaces, Applied General Topology 4 (2003), 255-261. [3] Lj.D.R. Koˇ cinac The Reznichenko property and the Pytkeev property in hyperspaces, Acta Math. Hungar. 107 (2005), 225-233. [4] Lj.D.R. Koˇ cinac, M. Scheepers, Function spaces and a property of Reznichenko, Topology Appl. 123 (2002), 135-143. [5] Lj.D.R. Koˇ cinac, M. Scheepers, Combinatorics of open covers (VII): Groupability, Fund. Math. 179 (2003), 131-155. Vladimir Pavlovi´ c

December 14, 2005


25

[6] V.I. Malyhin, G. Tironi Weakly Fr´ echet-Urysohn and Pytkeev spaces, Topology Appl. 104 (2000), 181-190. [7] V. Pavlovi´ c, Selectively stricty A-function spaces, East-West J. Math., accepted. [8] V. Pavlovi´ c, A selective version of the property of Reznichenko in function spaces, submitted. [9] M. Sakai, The Pytkeev property and the Reznichenko property in function spaces, Note di Matematica 22:2 (2003), 43-52.


December 14, 2005