Question about Topological Subspace definition Let $(S,\tau)$ be a topological space, and let $H\subseteq S$.
Using this definition, we can define the topological subspace $(H,τ_H)$ where $τ_H:=\{ U \cap H : U \in τ \}$.
Now, having just begun my independent study of topology, I found this definition to be a little surprising. Why not just define the subspace so that $τ_H:=\{U\in τ : U\subseteq H\}$?
 A: Recall that a topological space $(X,τ)$ must have the property that $X\inτ$. That is, $X$ itself must be "open."
The problem with defining $τ_H$ to be $\{U\in τ : U\subseteq H\}$ is that $(H,τ_Η)$ could only be a topology if $H$ is "open" in our original set $S$. We want a definition which allows us to take any (nonempty) subset of $S$ and form a topology.
A: Regarding your question (basically just saying the same as Pascal):
If $Y \not \in \tau$, then $Y \subset Y$ would not be open, which is an axiom for a topology. Therefore we have to define the subspace topology in a way which does not depend on properties of the subset.
The idea for the definition is the following:
If we have a topological space $(X,\tau)$ and a subset $Y \subset X$, then we want the inclusion map $i \colon Y \rightarrow X$ to be continuous. This means for every open set $U \subset X$ we need the preimage under the inclusion map $i^{-1}(U) = U \cap Y$ to be open in $Y$. We now take th smallest topology such that this is satisfied (which is just taking this as definition for open sets) and that is how the subspace topology is defined.
