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

Let $f_s$ and $f_t$ be tent maps on the unit interval. In this paper we give a new proof of the fact that if the critical points of $f_s$ and $f_t$ are periodic and the inverse limit spaces $(I,f_s)$ and $(I,f_t)$ are homeomorphic, then s = t. This theorem was first proved by Kailhofer. The new proof in this paper simplifies the proof of Kailhofer. Using the techniques of the paper we are also able to identify certain isotopies between homeomorphisms on the inverse limit space.

The main purpose of this paper is to show that any localic group is complete in its two-sided uniformity, settling a problem open since work began in this area a decade ago. In addition, a number of other results are established, providing in particular a new functor from topological to localic groups and an alternative characterization of $LT$-groups.

Let $𝐏$ be a porosity-like relation on a separable locally compact metric space $E$. We show that the $\sigma$-ideal of compact $\sigma$-$𝐏$-porous subsets of $E$ (under some general conditions on $𝐏$ and $E$) forms a ${\Pi }_{\mathbf{1}}^{\mathbf{1}}$-complete set in the hyperspace of all compact subsets of $E$, in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. It is shown in the cases of the $\sigma$-ideals of $\sigma$-porous sets, $\sigma$-$\langle g\rangle$-porous sets, $\sigma$-strongly porous sets, $\sigma$-symmetrically porous sets...

Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{\omega }$ is $F_{\sigma \delta }$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{\omega }$. We show that $S_{\omega }$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover,...

If $f:[0,1]\to ℝ$ is strictly increasing and continuous define $T_{f}x=f\left(x\right)(mod1)$. A transformation $\tilde{T}:[0,1]\to [0,1]$ is called $\epsilon$-close to $T_f$, if $\tilde{T}x=\tilde{f}\left(x\right)(mod1)$ for a strictly increasing and continuous function $\tilde{f}:[0,1]\to ℝ$ with $\parallel \tilde{f}{-f\parallel }_{\infty }<\epsilon$. It is proved that the topological pressure $p(T_f,g)$ is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the continuity of the maximal measure is shown, if a certain condition is satisfied. Then it is proved that the topological pressure is upper semi-continuous for every continuous function $g:[0,1]\to ℝ$, if and only if $0$ is...

Let $C$ be a convex subset of a complete convex metric space $X$, and $S$ and $T$ be two selfmappings on $C$. In this paper it is shown that if the sequence of Ishikawa iterations associated with $S$ and $T$ converges, then its limit point is the common fixed point of $S$ and $T$. This result extends and generalizes the corresponding results of Naimpally and Singh [6], Rhoades [7] and Hicks and Kubicek [3].

Let X be a nonempty set of cardinality at most $2^{\aleph ₀}$ and T be a selfmap of X. Our main theorem says that if each periodic point of T is a fixed point under T, and T has a fixed point, then there exist a metric d on X and a lower semicontinuous map ϕ :X→ ℝ ₊ such that d(x,Tx) ≤ ϕ(x) - ϕ(Tx) for all x∈ X, and (X,d) is separable. Assuming CH (the Continuum Hypothesis), we deduce that (X,d) is compact.