Equivalence of topological compactifications

Solution 1:

The main reason for defining the equivalence this way is to give an order structure not to the points in $X$, but to the different compactifications of $X$. You should start by defining an order on the compactifications by saying that one compactification $(Y_1,c_1)$ is less than or equal to another compactification $(Y_2,c_2)$ if and only if there is a continuous surjection $\phi:Y_2\rightarrow Y_1$ such that $c_2\circ \phi=c_1$. In this case the equivalent compactifications are the ones for which you have a homeomorphism. The reason behind wanting an order to the compactifications is to be able to then use Zorn's Lemma to obtain a maximal element in the family of compactifications of a space $X$, which is also known as the Stone-Čech compactification $\beta X$.

Solution 2:

This question is years old but I thought I'd share another point of view about the relevance of having a commutative diagram $f\circ c_1 = c_2$, for people in the future could perhaps find it useful too.

Using your notation, imagine that $Y_1$ and $Y_2$ are actually supersets of $X$ and assume that $c_i:X\hookrightarrow Y_i$ is the inclusion for $i\in\{1,2\}$. Then the compactifications $(Y_1,c_1)$ and $(Y_2,c_2)$ are equivalent iff there is a homeomorphism $f:Y_1\to Y_2$ such that $f(x)=x$ for all $x\in X$.

This says that $X$ is embedded in the same way in both compactifications.

In the general case where $Y_i$ is not necessarily a superset of $X$, the previous notion still makes sense, because the embedding $c_1$ acts essentially as a "generalized inclusion". The main idea of a compactification is to think of a space as if it were compact so that one can take advantage of properties that compact spaces have, so it seems natural to ask when two ways of compactifying a space $X$ end up looking exactly the same from the perspective of $X$.

David Ullrich's answer in this post is very insightful of the essence of this concept and the interpretation that I tried to provide. In his example, not only are his two compactifications not equivalent $-$ they are not even comparable (w.r.t. the preorder mentioned in Simon_Peterson's answer).