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.

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.

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 $Y$).