Geodesic
Degree of Discrete Generation of Compact Sets
Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 396-401.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper, under the continuum hypothesis, we construct an example of a discretely generated compact set $X$ whose square is not discretely generated. For each compact set $X$, there is an ordinally valued characteristic $\operatorname{idc}(X)$, which is the least number of iterations of the $d$-closure generating, as a result, the closure of any original subset $X$. We prove that if $\chi(X)\le\omega_\alpha$, then $\operatorname{idc}(X)\le\alpha+1$.
Keywords: discretely generated compact set, compact Hausdorff space, $d$-closure, compact extension, one-point compactification, continuum hypothesis. 
@article{MZM_2010_87_3_a6,
     author = {A. V. Ivanov and E. V. Osipov},
     title = {Degree of {Discrete} {Generation} of {Compact} {Sets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {396--401},
     publisher = {mathdoc},
     volume = {87},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a6/}
}
TY  - JOUR
AU  - A. V. Ivanov
AU  - E. V. Osipov
TI  - Degree of Discrete Generation of Compact Sets
JO  - Matematičeskie zametki
PY  - 2010
SP  - 396
EP  - 401
VL  - 87
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a6/
LA  - ru
ID  - MZM_2010_87_3_a6
ER  - 
%0 Journal Article
%A A. V. Ivanov
%A E. V. Osipov
%T Degree of Discrete Generation of Compact Sets
%J Matematičeskie zametki
%D 2010
%P 396-401
%V 87
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a6/
%G ru
%F MZM_2010_87_3_a6
A. V. Ivanov; E. V. Osipov. Degree of Discrete Generation of Compact Sets. Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 396-401. http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a6/

[1] A. Dow, M. G. Tkachenko, V. V. Tkachuk, R. G. Wilson, “Topologies generated by discrete subspaces”, Glas. Mat. Ser. III, 37:1 (2002), 187–210 | MR | Zbl

[2] V. V. Fedorchuk, “Bikompakt, vse beskonechnye zamknutye podmnozhestva kotorogo $n$-merny”, Matem. sb., 96:1 (1975), 41–62 | MR | Zbl