A proof of the C⁰-closing lemma for noninvertible discrete dynamical systems and its extension to the noncompact case are presented.

Let B(κ,λ) be the subalgebra of P(κ) generated by ${\left[\kappa \right]}^{\le \lambda }$. It is shown that if B is any homomorphic image of B(κ,λ) then either $\left|B\right|<2^{\lambda }$ or $\left|B\right|={\left|B\right|}^{\lambda }$; moreover, if X is the Stone space of B then either $\left|X\right|\le 2^{2^{\lambda }}$ or $\left|X\right|=\left|B\right|={\left|B\right|}^{\lambda }$. This implies the existence of 0-dimensional compact $T_2$ spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.

In this paper two new cardinal functions are introduced and investigated. In particular the following two theorems are proved: (i) If $X$ is a functionally Hausdorff space then $\left|X\right|\le 2^{fs\left(X\right){\psi }_{\tau }\left(X\right)}$; (ii) Let $X$ be a functionally Hausdorff space with $fs\left(X\right)\le \kappa$. Then there is a subset $S$ of $X$ such that $\left|S\right|\le 2^{\kappa }$ and $X=\bigcup \{c{l}_{\tau \theta }\left(A\right):A\in {\left[S\right]}^{\le \kappa }\}$.

The aim of this paper is to show, using the reflection principle, three new cardinal inequalities. These results improve some well-known bounds on the cardinality of Hausdorff spaces.

In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to Fedeli [2] and the third theorem of this paper is a common generalization to: (a) (Arhangel’skii [1]) If $X$ is a $T_1$ space such that (i) $L\left(X\right)t\left(X\right)\le \kappa$, (ii) $\psi \left(X\right)\le 2^{\kappa }$, and (iii) for all $A\in {\left[X\right]}^{\le 2^{\kappa }}$, $\left|\overline{A}\right|\le 2^{\kappa }$, then $\left|X\right|\le 2^{\kappa }$; and (b) (Fedeli [2]) If $X$ is a $T_2$-space then $\left|X\right|\le 2^{aql\left(X\right)t\left(X\right){\psi }_c\left(X\right)}$.

Two variations of Arhangelskii’s inequality $$\left|X\right|⩽2^{\chi \left(X\right)-L\left(X\right)}$$ for Hausdorff X [Arhangel’skii A.V., The power of bicompacta with first axiom of countability, Dokl. Akad. Nauk SSSR, 1969, 187, 967–970 (in Russian)] given in [Stavrova D.N., Separation pseudocharacter and the cardinality of topological spaces, Topology Proc., 2000, 25(Summer), 333–343] are extended to the classes with finite Urysohn number or finite Hausdorff number.

We prove that the cardinality of power homogeneous Hausdorff spaces X is bounded by $d{\left(X\right)}^{\pi \chi \left(X\right)}$. This inequality improves many known results and it also solves a question by J. van Mill. We further introduce Δ-power homogeneity, which leads to a new proof of van Douwen’s theorem.

We introduce the cardinal invariant $\theta$-$a{L}^{\text{'}}\left(X\right)$, related to $\theta$-$aL\left(X\right)$, and show that if $X$ is Urysohn, then $\left|X\right|\le 2^{\theta \text{-}a{L}^{\text{'}}\left(X\right)\chi \left(X\right)}$. As $\theta$-$a{L}^{\text{'}}\left(X\right)\le aL\left(X\right)$, this represents an improvement of the Bella-Cammaroto inequality. We also introduce the classes of firmly Urysohn spaces, related to Urysohn spaces, strongly semiregular spaces, related to semiregular spaces, and weakly $H$-closed spaces, related to $H$-closed spaces.