Does the function $f(x)=x^2$ admit a continuous extension $\widetilde{f}:\beta\mathbb{R}\longrightarrow\mathbb{R}$ to the Stone-Cech compactification? Proof. If $f$ admitted a continuous extension to the Stone-Cech compactification, then the triangle/maps would commute. The triangle being: $f =...

Can someone explain, given a set of disjoint open intervals, does the 1 point compactification look more like: Or Also how do we know that the compactification would necessarily be in a higher dimension?

https://en.wikipedia.org/wiki/Alexandroff_extension Definition: one point compactification Let $X$ be any topological space, and let $ \infty$ be any object which is not already an element of $X$. Put $ X^{*}=X\cup \{\infty \}$, and topologize $X^* $ by taking as open sets all the o...

The tag compactification was created a few days ago. This tag has been created and removed before. At the moment there are 4 questions with this tag. Somewhat related tag was discussed here: Would tags such as "ultrafilters" or "Stone-Cech compactification" be too specific? After that discussio...