## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Observations on spaces with property $(DC(\omega_1))$

#### Abstract

A topological space $X$ has property $(DC(\omega_1))$ if it has a dense subspace every uncountable subset of which has a limit point in $X$. In this paper, we make some observations on spaces with property $(DC(\omega_1))$. In particular, we prove that the cardinality of a space $X$ with property $(DC(\omega_1))$ does not exceed $\mathfrak c$ if $X$ satisfies one of the following conditions: (1) $X$ is normal and has a rank $2$-diagonal; (2) $X$ is perfect and has a rank $2$-diagonal; (3) $X$ has a rank $3$-diagonal; (4) $X$ is perfect and has countable tightness. We also prove that if $X$ is a regular space with a $G_\delta$-diagonal and property $(DC(\omega_1))$ then the cardinality of $X$ is at most $2^\mathfrak c$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 25, Number 1 (2018), 55-62.

Dates
First available in Project Euclid: 11 April 2018

https://projecteuclid.org/euclid.bbms/1523412052

Digital Object Identifier
doi:10.36045/bbms/1523412052

Mathematical Reviews number (MathSciNet)
MR3784505

Zentralblatt MATH identifier
06882541

Subjects
Primary: 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.)
Secondary: 54E35: Metric spaces, metrizability

#### Citation

Xuan, Wei-Feng; Shi, Wei-Xue. Observations on spaces with property $(DC(\omega_1))$. Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 1, 55--62. doi:10.36045/bbms/1523412052. https://projecteuclid.org/euclid.bbms/1523412052