The characterizations of dense-pseudocompact and dense-connected spaces. (English) Zbl 07695411

Summary: Assume that \(\mathcal{P}\) is a topological property of a space \(X\), then we say that \(X\) is dense-\(\mathcal{P}\) if each dense subset of \(X\) has the property \(\mathcal{P} \). In this paper, we mainly discuss dense subsets of a space \(X\), and we prove that:
(1) if \(X\) is Tychonoff space, then \(X\) is dense-pseudocompact if and only if the range of each continuous real-valued function \(f\) on \(X\) is finite, if and only if \(X\) is finite, if and only if \(X\) is hereditarily pseudocompact;
(2) \(X\) is dense-connected if and only if \(\overline{U} = X\) for any non-empty open subset \(U\) of \(X\);
(3) \(X\) is dense-ultraconnected if and only if for point \(x \in X\), we have \(\overline{\{ x \}} = X\) or \(\{x \} \cup(X \setminus \overline{\{ x \}})\) is the unique open neighborhood of \(x\) in \(\{x \} \cup(X \setminus \overline{\{ x \}})\), if and only if for any two points \(x\) and \(y\) in \(X\), we have \(x \in \overline{\{ y \}}\) or \(y \in \overline{\{ x \}} \).
Moreover, we give a characterization of a topological group (resp., paratopological group, quasi-topological group) \(G\) such that \(G\) is dense-connected.


22A05 Structure of general topological groups
54B05 Subspaces in general topology
54C30 Real-valued functions in general topology
54D05 Connected and locally connected spaces (general aspects)
54H11 Topological groups (topological aspects)
Full Text: DOI arXiv


[1] Arhangel’skiı̌, A.; Tkachenko, M., Topological Groups and Related Structures (2008), Atlantis Press, Paris, World Scientific Publishing Co. Pte. Ltd.: Atlantis Press, Paris, World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1323.22001
[2] Banakh, T.; StelmakhT, Y., Examples of strongly rigid countable (semi) Hausdorff spaces
[3] Brown, M., A countable connected Hausdorff space, Bull. Am. Math. Soc., 59, 367 (1953)
[4] Comfort, W. W.; Robertson, L. C., Proper pseudocompact extensions of compact Abelian group topologies, Proc. Am. Math. Soc., 86, 1, 173-178 (1982) · Zbl 0508.22002
[5] Comfort, W. W.; Robertson, L. C., Extremal phenomena in certain classes of totally bounded groups, Diss. Math., 272, 1-42 (1988) · Zbl 0703.22002
[6] Comfort, W. W.; Van Mill, J., Extremal pseudocompact abelian groups are compact metrizable, Proc. Am. Math. Soc., 135, 12, 4039-4044 (2007) · Zbl 1138.22002
[7] Comfort, W. W.; van Mill, J., Concerning connected, pseudocompact Abelian groups, Topol. Appl., 33, 1, 21-45 (1989) · Zbl 0698.54003
[8] Dow, A.; Juhász, I., Dense k-separable compacta are densely separable, Topol. Appl., 283, Article 107351 pp. (2020) · Zbl 1461.54021
[9] Dikranjan, D.; Shakhmatov, D., A complete solution of Markov’s problem on connected group topologies, Adv. Math., 286, 286-307 (2016) · Zbl 1331.22003
[10] Dikranjan, D., The gentle, generous giant tampering with dense subgroups of topological groups, Topol. Appl., 259, 6-27 (2019) · Zbl 1414.22004
[11] Engelking, R., General Topology (1989), PWN: PWN Warzawa · Zbl 0684.54001
[12] Juhász, I.; Shelah, S., \( \pi(X) = \delta(X)\) for compact X, Topol. Appl., 32, 289-294 (1989) · Zbl 0688.54002
[13] Levy, R.; McDowell, R. H., Dense subsets of βX, Proc. Am. Math. Soc., 50, 1, 426-430 (1975) · Zbl 0313.54025
[14] Lin, F.; Wu, Q. Y.; Liu, C., Dense-separable groups and its applications in d-independence
[15] Maehara, R., On a connected dense proper subgroup of \(\mathbb{R}^2\) whose complement is connected, Proc. Am. Math. Soc., 97, 3, 556-558 (1986) · Zbl 0593.54037
[16] Wilcox, H. J., Dense subgroups of compact groups, Proc. Am. Math. Soc., 28, 2, 578-580 (1971) · Zbl 0215.40503
[17] Weston, J. H.; Shilleto, J., Cardinalities of dense sets, Topol. Appl., 6, 227-240 (1976) · Zbl 0321.54003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.