A confusion about open/closed sets Suppose $(X,\tau_X)$ is a topological space, and $Y \subset X$ a subset. We can equip $Y$ with the topology $\tau_Y:= \{Y \cap A: A\in \tau_X \}$, called the subspace topology. 
When one says "$W \subset Y$ is open", does it mean $W\in \tau_X$ and $W\subset Y$ or does it mean $W\in \tau_Y$?   
 A: It means $W=V\cap Y$ where $V\in \tau_X$.
A: It means the latter: one can have sets which are open in the subspace topology, but aren't in the topology of the bigger space.
Example: Consider the normal topology on $\mathbb{R}$, and consider the subspace topology on $[0,1]$. Clearly, $(0,1]$ is open in $[0,1]$, since $(0,1] = (0,2) \cap [0,1]$, but $(0,1]$ is not open in $\mathbb{R}$.
To use your notation, $W = (0,1]$ does not belong to $\tau_X$, hence does not satisfy the first definition.
A: It means the latter. The subspace topology is a surprisingly slippery concept for beginning students of topology to grasp. Like linear subspaces of vector spaces in linear algebra, a topological subspace $A\subseteq X$ of a topological space X is itself a topological space whose topology is induced by the larger topology,but is a different unique topology on A. 
Here's a simple example. Consider the real line with the usual metric topology. Consider the interval (-$\infty$,1] as a subspace of the real line. Now consider the interval (0,1]. Is this interval an open subspace in $\mathbb R$ ? Under the usual topology, no. But is it relatively open in the subspace (-$\infty$,1]? YES, because (0,1]= (-$\infty$,1]$\cap$ (0,2) and by definition, this lies in the subspace topology on
 (-$\infty$,1]! 
The best way to understand the subspace topology is to study and construct a lot of examples until you get really comfortable with the concept. 
