I'm trying to understand how the inclusion map can be used in a helpful way. What is its purpose in proofs?
In the context of topological spaces, how is it useful to include a set A into a superset B of A? Isn't the concept of embedding more useful? My understanding is from that perspective we can consider the inclusion map as trivial.
An explicit example would be helpful. Cheers
E.g. in topology there is the useful notion of an initial topology on a set: a topology on a set $X$ that is defined by there being a family of functions $f:X \to Y_i$, where the $Y_i$ already have a topology. A topology on $X$ can then be defined as the minimal one on $X$ so that all $f_i$ become continuous functions between $X$ (in that topology) and $Y_i$. I explain the theory here in more detail.
The subspace topology of a subset $A$ of a space $X$ is just the initial topology on $A$ for the one-element family $1_A:A \to X$, the inclusion mapping.
So defining it (it's a quite trivial map, but a natural one) allows us to put the subspace topology in the theory we have on initial topologies (universal mapping theorems, transitive law etc.; also links with category theory) and this gives us easy facts like the restriction of a continuous map is still continuous on the subspace (as $f\restriction_A = f \circ 1_A$ when $f: X \to Y$ is continuous), by applying the already known "compositions of continuous functions are continuous" fact.
So it's easier for theory and theorem-proving and that's what maths is about.
Embeddings are also examples of initial topologies: $f: X \to Y$ is an embedding iff $f$ is 1-1 and $X$ has the initial topology w.r.t. $f$. So at some abstract level, the inclusion map and embeddings are "the same".
