# Article

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Keywords:
cardinality; Discrete Countable Chain Condition; normal space; rank 2-diagonal; $G_\delta$-diagonal
Summary:
A topological space $X$ has a rank 2-diagonal if there exists a diagonal sequence on $X$ of rank $2$, that is, there is a countable family $\{\mathcal U_n\colon n\in \omega \}$ of open covers of $X$ such that for each $x \in X$, $\{x\}=\bigcap \{{\rm St}^2(x, \mathcal U_n)\colon n \in \omega \}$. We say that a space $X$ satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. We mainly prove that if $X$ is a DCCC normal space with a rank 2-diagonal, then the cardinality of $X$ is at most $\mathfrak c$. Moreover, we prove that if $X$ is a first countable DCCC normal space and has a $G_\delta$-diagonal, then the cardinality of $X$ is at most $\mathfrak c$.
References:
[1] Arhangel'skii, A. V., Buzyakova, R. Z.: The rank of the diagonal and submetrizability. Commentat. Math. Univ. Carol. 47 (2006), 585-597. MR 2337413 | Zbl 1150.54335
[2] Buzyakova, R. Z.: Cardinalities of ccc-spaces with regular $G_\delta$-diagonals. Topology Appl. 153 (2006), 1696-1698. DOI 10.1016/j.topol.2005.06.004 | MR 2227022 | Zbl 1094.54001
[3] Engelking, R.: General Topology. Sigma Series in Pure Mathematics 6 Heldermann, Berlin (1989). MR 1039321 | Zbl 0684.54001
[4] Ginsburg, J., Woods, R. G.: A cardinal inequality for topological spaces involving closed discrete sets. Proc. Am. Math. Soc. 64 (1977), 357-360. DOI 10.1090/S0002-9939-1977-0461407-7 | MR 0461407 | Zbl 0398.54002
[5] Hodel, R.: Cardinal functions I. Handbook of Set-Theoretic Topology North-Holland VII, Amsterdam K. Kunen et al. 1-61 North-Holland, Amsterdam (1984). MR 0776620 | Zbl 0559.54003
[6] Matveev, M.: A survey on star covering properties. Topology Atlas (1998), http://at.yorku.ca/v/a/a/a/19.htm
[7] Shakhmatov, D. B.: No upper bound for cardinalities of Tychonoff C.C.C. spaces with a $G_\delta$-diagonal exists. Commentat. Math. Univ. Carol. 25 (1984), 731-746. MR 0782022
[8] Uspenskij, V. V.: A large $F_\sigma$-discrete Frechet space having the Souslin property. Commentat. Math. Univ. Carol. 25 (1984), 257-260. MR 0768812
[9] Wiscamb, M. R.: The discrete countable chain condition. Proc. Am. Math. Soc. 23 (1969), 608-612. DOI 10.1090/S0002-9939-1969-0248744-1 | MR 0248744 | Zbl 0184.26304
[10] Xuan, W. F., Shi, W. X.: A note on spaces with a rank 3-diagonal. Bull. Aust. Math. Soc. 90 (2014), 521-524. DOI 10.1017/S0004972714000318 | MR 3270766 | Zbl 1305.54036
[11] Xuan, W. F., Shi, W. X.: A note on spaces with a rank 2-diagonal. Bull. Aust. Math. Soc. 90 (2014), 141-143. DOI 10.1017/S0004972713001184 | MR 3227139

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