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For $k\in \mathbb{N}\cup \left\{\infty \right\}$ and $U$ open in $ℝ$, let ${\mathrm{C}}^k\left(U\right)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f\in {\mathrm{C}}^k\left(U\right)$ there is $g\in {\mathrm{C}}^{\infty }\left(ℝ\right)$ with $U\subseteq \mathrm{coz}g$ and $h\in {\mathrm{C}}^k\left(ℝ\right)$ with ${fg|}_U{=h|}_U$. The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$. The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring ${\mathrm{C}}^k\left(ℝ\right)$ are deduced from this, in particular that ${\mathrm{Q}}_{\mathrm{cl}}\left({\mathrm{C}}^k\left(ℝ\right)\right)=\mathrm{Q}\left({\mathrm{C}}^k\left(ℝ\right)\right)$.

We raise the question of when a simplicial object in a catetgory is deemed contractible. The literature offers three definitions. One is the existence of an “extra degeneracy”, indexed by $-1$, which does not quite live up to the name. This can be strengthened to a “strong extra degeneracy". Another possibility is that it be homotopic to a constant simplicial object. Despite claims in the literature to the contrary, we show that all three are distinct concepts with strong extra degeneracy implies extra...

The set of isolated points (resp. $P$-points) of a Tychonoff space $X$ is denoted by $Is\left(X\right)$ (resp. $P\left(X\right))$. Recall that $X$ is said to be if $Is\left(A\right)\ne ⌀$ whenever $⌀\ne A\subset X$. If instead we require only that $P\left(A\right)$ has nonempty interior whenever $⌀\ne A\subset X$, we say that $X$ is . Many theorems about scattered spaces hold or have analogs for spaces. For example, the union of a locally finite collection of SP-scattered spaces is SP-scattered. Some known theorems about Lindelöf or paracompact scattered spaces hold also in case the spaces are SP-scattered....