What does it mean that "compactification is defined only with respect to the topology of the base space"? I am reading about topological compactifications, one of the materials I came accross is this paper by Benjamin Vejnar.
My question:
What does it mean that "the compactification is defined only
with respect to the topology of the base space and does not depend on the concrete representation of the space"?
I dont know how interpret it. Maybe my problem is that I don't know how are compactifications and topological compactifications usually defined (I have seen construction of Stone-Čech and Alexandroff one-point compactifciation, but not many other examples). Thank you for any insights!
Definition: Topological compactification (or "H-compactification") is such compactification of a topological space such that all autohomeomorphisms on that space can be continuously extended into autohomeomorphisms of the compactification.
Disclaimer: "Topological compactification" is really NOT every compactification. I have already tried to explain this in a different question but didn't get any answer.
 A: Like many brief and cryptic sentences found in introductions of research papers, this "concrete representation" sentence is most likely intended to be an appeal to some kind of intuition which a less fortunate reader may not share. In this case, it is an appeal to an intuitive understanding of a broad class of compactification constructions.
There are many, many, many ways to construct compactifications by embedding $X$ into compact spaces: If $f : X \to C$ is any embedding of $X$ into a compact space $C$, meaning a homeomorphism from $X$ onto a subspace of $C$, then the closure of the image $\overline{\text{image}(f)}$ may be regarded as a compactification of $X$.
Let's take $X = (0,1]$ for example.
We could embed $f : (0,1] \to \mathbb R^2$ by the formula
$$f(x) = (x,\sin(1/x))
$$
The subspace $\text{image}(f) \subset \mathbb R^2$ is bounded and is therefore contained in a closed ball, which is compact. The closure $\overline{\text{image}(f)}$ is therefore a compactification of $(0,1]$, known as the topologist's sine curve (or most of the topologist's sine curve, at least, including the most interesting portion of it). Notice that $\overline{\text{image}(f)}$ is obtained by adding a line segment to $\text{image}(f)$, namely $\{0\} \times [-1,+1]$.
Or we could choose a different embedding $f : (0,1] \to \mathbb R^2$ using the formula
$$f(x) = (r(x) \cos(\theta(x)), r(x) \sin(\theta(x)))
$$
where $r(x)= 1 - x$ and $\theta(x) = \frac{1}{x}$. Again $\text{image}(f)$ is bounded, which gives a compactification $\overline{\text{image}(f)}$ that is obtained from $\text{image}(f)$ by adding the unit circle to $\text{image}(f)$.
Now let your imagination run wild: by choosing "concrete representations" of the space $X = (0,1]$, for example representations as bounded subsets of a Euclidean space $\mathbb R^n$, we obtain many, many strange compactifications of $X$.
What that sentence means, in a rather rough and intuitive sense, is this: an $H$-compactification is not one of these random, silly, "concrete representation" compactifications.
A: The usual definition of a compactification $(Y,e)$ of a space $X$ is pair of a compact space $Y$ and a continuous $e: X \to Y$ so that $e[X]$ is dense in $Y$ and $e\restriction X \to e[X]$ is a homeomorphism.
Often we take $Y$ to be Hausdorff and stipulate $X$ to be non-compact to avoid some trivialities.
In practice we pretend $X \subseteq Y$ as a subspace (as $e$ is an embedding anyway) and $X$ is dense, and write $Y$ as $\gamma X$ or $\alpha X$ or $\beta X$ (if we have some construction of sorts to go from $X$ to $Y$).
