The problem whether every topological space $X$ has a compactification $Y$ such that every continuous mapping $f$ from $X$ into a compact space $Z$ has a continuous extension from $Y$ into $Z$ is answered in the negative. For some spaces $X$ such compactifications exist.
@article{118481, author = {Miroslav Hu\v sek}, title = {\v Cech-Stone-like compactifications for general topological spaces}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {33}, year = {1992}, pages = {159-163}, zbl = {0754.54014}, mrnumber = {1173757}, language = {en}, url = {http://dml.mathdoc.fr/item/118481} }
Hušek, Miroslav. Čech-Stone-like compactifications for general topological spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) pp. 159-163. http://gdmltest.u-ga.fr/item/118481/
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