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?

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. 
