We introduce the properties of a space to be strictly $WFU\left(M\right)$ or strictly $SFU\left(M\right)$, where $\varnothing \ne M\subset {\omega }^{*}$, and we analyze them and other generalizations of $p$-sequentiality ($p\in {\omega }^{*}$) in Function Spaces, such as Kombarov’s weakly and strongly $M$-sequentiality, and Kocinac’s $WFU\left(M\right)$ and $SFU\left(M\right)$-properties. We characterize these in $C_{\pi }\left(X\right)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $WFU\left(L\right(M\left)\right)$-property, where $L\left(M\right)=\{{}^{\lambda }p:\lambda <{\omega }_1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset {\omega }^{*}$; and that $C_{\pi }\left(X\right)$ is a $WFU\left(L\right(M\left)\right)$-space iff...

Given a free ultrafilter $p$ on $\mathbb{N}$ and a space $X$, we say that $x\in X$ is the $p$-limit point of a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in $X$ (in symbols, $x=p$-${lim}_{n\to \infty }{x}_n$) if for every neighborhood $V$ of $x$, $\{n\in \mathbb{N}:x_n\in V\}\in p$. By using $p$-limit points from a suitable metric space, we characterize the selective ultrafilters on $\mathbb{N}$ and the $P$-points of ${\mathbb{N}}^{*}=\beta \left(\mathbb{N}\right)\setminus \mathbb{N}$. In this paper, we only consider dynamical systems $(X,f)$, where $X$ is a compact metric space. For a free ultrafilter $p$ on ${\mathbb{N}}^{*}$, the function $f^p:X\to X$ is defined by $f^p\left(x\right)=p$-${lim}_{n\to \infty }{f}^n\left(x\right)$ for each $x\in X$. These functions are not continuous in general....

The canonization theorem says that for given $m,n$ for some $m^{*}$ (the first one is called $ER(n;m)$) we have for every function $f$ with domain ${\left[1,\cdots ,m^{*}\right]}^n$, for some $A\in {\left[1,\cdots ,m^{*}\right]}^m$, the question of when the equality $f\left(i_1,\cdots ,i_n\right)=f\left(j_1,\cdots ,j_n\right)$ (where $i_1<\cdots <i_n$ and $j_1<\cdots j_n$ are from $A$) holds has the simplest answer: for some $v\subseteq \{1,\cdots ,n\}$ the equality holds iff ${\bigwedge }_{\ell \in v}{i}_{\ell }=j_{\ell }$. We improve the bound on $ER(n,m)$ so that fixing $n$ the number of exponentiation needed to calculate $ER(n,m)$ is best possible.