Purpose of the inclusion map 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 
 A: 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". 
