%0 Journal Article
%T Connectifying a topological space by adding one point
%J Journal of the Iranian Mathematical Society
%I Iranian Mathematical Society
%Z 2717-1612
%A Koushesh, M. R.
%D 2021
%\ 06/01/2021
%V 2
%N 2
%P 81-110
%! Connectifying a topological space by adding one point
%K One-point connectification
%K one-point compactification
%K Stone-Cech compactification
%K Local connectedness
%K Component
%R 10.30504/jims.2022.321037.1050
%X P. Alexandroff proved that a locally compact $T_2$-space has a $T_2$ one-point compactification (obtained by adding a ``point at infinity'') if and only if it is non-compact. Also he asked for characterizations of spaces which have one-point connectifications. Here, we study one-point connectifications, and in an attempt to answer Alexandroff's question (and in analogy with Alexandroff's theorem) we prove that in the class of $T_i$-spaces ($i=3\frac{1}{2},4,5$) a locally connected space has a one-point connectification if and only if it has no compact component. We extend this theorem to the case $i=6$ by assuming the set-theoretic assumption $\mathbf{MA}+\neg\mathbf{CH}$, and to the case $i=2$ by slightly modifying its statement. We further extended the theorem by proving that a locally connected metrizable (resp. paracompact) space has a metrizable (resp. paracompact) one-point connectification if and only if it has no compact component. Contrary to the case of the one-point compactification, a one-point connectification, if exists, may not be unique. We instead consider the collection of all one-point connectifications of a locally connected locally compact space in the class of $T_i$-spaces ($i=3\frac{1}{2},4,5$). We prove that this collection, naturally partially ordered, is a compact conditionally complete lattice whose order structure determines the topology of all Stone--$\rm\check{C}$ech remainders of components of the space.
%U http://jims.ims.ir/article_147063_b75fa640bd903974143bcd38d88c197f.pdf