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The class of Hausdorff spaces (or of Tychonoff spaces) which admit a Hausdorff (respectively Tychonoff) sequentially compact extension has not been characterized. We give some new conditions, in particular, we prove that every Tychonoff locally sequentially compact space has a Tychonoff one-point sequentially compact extension. We also give some results about extension of functions and about lattice properties of the family of all minimal sequentially compact extensions of a given space.

Let $F\subset C^{\ast }(X)$ be a vector sublattice over $ℝ$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_z$ containing $F$ such that $\omega (Z(F))=\omega (Z(F_z))=e_{F}X$. In particular this implies that $\omega (Z(F))\ge e_{F}X$. Conditions in order to be $\omega (Z(F))=e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $C{l}_{\alpha X}(\alpha X\setminus X)$ is $0$-dimensional, then there is an algebra $A\subset C^{ast}(X)$ such...

If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$, then $f(x)$ has at least one root a with the following property: if $\mu \le k\le n$, where $\mu$ is the multiplicity of $\alpha$, then $f^{(k)}(\alpha )\ne 0$ (such a root is said to be a "free" root of $f(x)$). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$) with coefficients in a field of positive characteristic $p>n$ (Sudbery's Conjecture). In this paper it is shown that,...

If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$, then $f(x)$ has at least one root a with the following property: if $\mu \le k\le n$, where $\mu$ is the multiplicity of $\alpha$, then $f^{(k)}(\alpha )\ne 0$ (such a root is said to be a "free" root of $f(x)$). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$) with coefficients in a field of positive characteristic $p>n$ (Sudbery's Conjecture). In this paper it is shown that,...

Let $F\subset C^{\ast }(X)$ be a vector sublattice over $ℝ$ which separates points from closed sets of $X$. The compactification $e_{F}X$ obtained by embedding $X$ in a real cube via the diagonal map, is different, in general, from the Wallman compactification $\omega (Z(F))$. In this paper, it is shown that there exists a lattice $F_z$ containing $F$ such that $\omega (Z(F))=\omega (Z(F_z))=e_{F}X$. In particular this implies that $\omega (Z(F))\ge e_{F}X$. Conditions in order to be $\omega (Z(F))=e_{F}X$ are given. Finally we prove that, if $\alpha X$ is a compactification of $X$ such that $C{l}_{\alpha X}(\alpha X\setminus X)$ is $0$-dimensional, then there is an algebra $A\subset C^{ast}(X)$ such...

In the theory of compactifications, Magill's theorem that the continuous image of a remainder of a space is again a remainder is one of the most important theorems in the field. It is somewhat unfortunate that the theorem holds only in locally compact spaces. In fact, if all continuous images of a remainder are again remainders, then the space must be locally compact. This paper is a modification of Magill's result to more general spaces. This of course requires restrictions on the nature of the...