Difference between subspace and subset in topology

Question

In Murkres's book Topology, I read that Y(a subset of some set X）is called the subspace of X if we consider the subspace topology. But what is the difference here between being a subspace and a subset? the definition in the book

Answer 1

You could ask the same question about topological spaces as a whole: What is the difference between a set and a topological space? The answer there is quite clear: A topological space is a set together with a topology on that set.

Your question has a quite similar but more specific answer: a subspace of a topological space $X$ is a subset $Y \subset X$ together with a specific topology on $Y$, namely the subspace topology.

Answer 2

Part1

This has been sorta asked recently for vector spaces. (Lee Mosher answered there too.) The analogy I'd like to use here is subgroup vs subset group:

You can make a group $G$ of the set $[0,2\pi)$ and some operation (I forgot the specifics, but I believe it's described in this elementary introduction of adding angles) s.t. $G$ is isomorphic to the circle group. $G$ then is not a subgroup of $\mathbb R$. But since $G$'s set $[0,2\pi)$ is a subset of $\mathbb R$ and since both $G$ and $\mathbb R$ are groups, I like to think of $G$ as a 'group subset' of $\mathbb R$: The sets can be made into groups with 1 a subset of the other, but the subset group's structure is not inherited from the superset group's structure.

Part2

Going back to topology:

Just pick any non-standard topology $\mathscr T$ you want for $\mathbb R$ to make $(\mathbb R,\mathscr T)$. Then for standard topology $\mathscr S$ on $\mathbb R$, we have that $(\mathbb R,\mathscr T)$ and $(\mathbb R,\mathscr S)$ are subsets (or topological subsets or whatever) of each other but they are not necessarily topological subspaces of each other.

Part3

Notice that the definition in the book for subspace topology says that with this topology. If you have $X$ and $Y$ as just sets, then...

1. $\mathscr T$ is a topology on $X$ and then $\mathscr T_Y$ is a topology on $Y$.

2. Then $(Y, \mathscr T_Y)$ and $(X,\mathscr T)$ are topological spaces.

3. Then $(Y, \mathscr T_Y)$ is then defined as a topological subspace of $(X,\mathscr T)$ in the sense that $Y$ is a subset of $X$ and that the topology $\mathscr T_Y$ on $Y$ is inherited from the topology $\mathscr T$ on $X$.

You can come up with a different topology $\mathscr A$ on $Y$. In this case $(Y, \mathscr A)$ is not (necessarily) a topological subspace of $(X,\mathscr T)$.

Part4

The point of this topology for subsets into becoming subspaces is like...

If we're talking about say continuity of some map $f:(X,\mathscr T) \to (Z,\mathscr B)$ and then we later restrict the map $f|_Y:(Y,\text{what topology?}) \to (Z,\mathscr B)$ to say $f|_Y$ is 'still continuous', then for this to be sensible we must have the topological structure on $Y$ for the map $f|_Y$ is $\mathscr T_Y$.

Part5

Also just now I saw this related question: What does it mean to equip a subset with the subspace topology?