Topology of subspace Suppose that $X$ is a topological space and that $Y$ is a subset of $X$.
A subset $V$ of the set $Y$ is said to be open in the space $Y$ when there exists an open subset $U$ of the space $X$ such that $V = Y \cap U$. 
Let's suppose that I encounter a space $Y$ that is a subset of another space $X$. Is the space $Y$ automatically equipped with the subspace topology defined in the previous paragraph? What is stopping me from claiming that the space $Y$ has some other topology, like the indiscrete topology?
 A: $Y$ need not have the same topology as that which it would inherit from $X$.  As a simple and fairly important example, take the real numbers with the usual topology.  Because of considerations like we're about to discuss, we don't just talk about $\mathbb{R}$ but instead about the set and the space as two different objects, $(\mathbb{R}, \mathcal{T}_u)$.  Now think about the integers.  One could allow them to inherit the subspace topology.  However, suppose you are a number theorist and for technical reasons, you would much rather have a different topology--namely, you are much more turned on by the topology of $p$-adic numbers.  Then the topology you put on the naturals is not the same one they'd inherit from the usual topology on the reals.
It's better not to think of encountering topologies on your walk through the woods.  Sure, mathematicians have charted these woods and you're following in their footsteps, so you are encountering what they've mapped.  But a better analogy is that, when you have a set, you put a topology on it, as a matter of choice.  You could keep the underlying set the same, and change your mind about which topology you'll put on it, depending on what your goals are.
A: If $(X, \mathcal{T})$ is a topological space and $Y$ is a subset of $X$, then by default (if nothing is said), $Y$ is seen as a topological space with the subspace topology induced from $\mathcal{T}$. 
This is similar to when we have a group $(G,\cdot)$, and we have $H \subseteq G$, we wonder whether $H$ is a group under the same operation as $G$ has, and get a notion of subgroup. It's a general phenomenon in maths, to consider substructures of a certain structure as having a structure as closely related to the large one as possible. 
