#### Glasnik Matematicki, Vol. 43, No.1 (2008), 205-217.

### ONE MORE VARIATION OF THE POINT-OPEN GAME

### Arcos Daniel Jardón and Vladimir V. Tkachuk

Departamento de Matemáticas, Universidad Autónoma Metropolitana,
San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa, C.P. 09340, México

*e-mail:* `jardon60@hotmail.com`

*e-mail:* `vova@xanum.uam.mx`

**Abstract.** A topological game "Dense
*G*_{δσ}-sets" (also denoted by
DG)
is introduced as follows: for any
*n* ω at the *n*-th
move the player *I* takes a point
*x*_{n} v *X* and
*II* responds by taking a *G*_{δ}-set *Q*_{n} in the
space *X* such that *x*_{n}
*Q*_{n}. The play stops after
ω moves and *I* wins if the set
{*Q*_{n} :
*n* ω}
is dense in *X*. Otherwise the player *II* is declared to be the winner.
We study classes of spaces on
which the player *I* has a winning strategy. It is evident
that the *I*-favorable spaces constitute a generalization of
the class of separable spaces. We show that there exists a
neutral space for the game
DG
and prove, among
other things, that Lindelöf scattered spaces and dyadic spaces
are *I*-favorable. We characterize *I*-favorability for the game
DG
in the spaces
*C*_{p}(*X*); one of the applications is that, for a Lindelöf
Σ-space *X*, the space *C*_{p}(*X*) is *I*-favorable for
DG
if and only if *X* is
ω-monolithic.

**2000 Mathematics Subject Classification.**
54H11, 54C10, 22A05, 54D25, 54C25.

**Key words and phrases.** Topological game, player, winning strategy,
dense *G*_{δσ}-sets, separable space, dyadic
compact space, scattered compact space, neutral space, function
space.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.43.1.14

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