#### Glasnik Matematicki, Vol. 37, No.1 (2002), 187-210.

### TOPOLOGIES GENERATED BY DISCRETE SUBSPACES

### A. Dow, M. G. Tkachenko, V. V. Tkachuk and
R. G. Wilson

Department of Mathematics and Statistics, York University,
4700 Keele Street, North York, Ontario, Canada M3J 1P3

*e-mail:* `dowa@mathstat.yorku.ca`
Departamento de Matematicas, Universidad Autonoma Metropolitana,
Av. San Rafael Atlixco 486, Col. Vicentina, Iztapalapa,
C. P. 09340 Mexico, D.F.

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

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

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

**Abstract.** A topological space *X* is called
discretely generated if for every subset *A*
⊂ *X*
we have

*A* =
∪
{*D* :
*D*
⊂ *A*
and *D* is a discrete subspace of *X*}.

We say that *X* is weakly discretely generated if
*A*
⊂ *X*
and *A* ≠
*A*
implies *D* \
*A* ≠ Ø
for some discrete *D*
⊂ *A*.
It is established that sequential spaces, monotonically normal
spaces and compact countably tight spaces are discretely generated.
We also prove taht every compact space is weakly discretely
generated and under the Continuum Hypothesis any dyadic
discretely generated space is metrizable.
**2000 Mathematics Subject Classification.**
54H11, 54C10, 22A05, 54D25, 54C25.

**Key words and phrases.** Discretely generated space,
weakly discretly generated space, dyadic space, countably
tight space.

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