If $X $ is a topological space and $ Y $ is a subspace of $ X$, then what does it mean when we say $Y$ is a path connected subspace of $X$?
Does it mean that any two points $ x $ and $ y $ in $ Y $ has a path in $Y$ (like if $X $ is path connected then any two points $ x $ and $y $ in $X$ has a path in $X$) ?
It means that it is path-connected if we consider in $Y$ the subspace topology. This is the same thing as asserting that, if $x,y\in Y$, the there is a continuous map $\gamma\colon[0,1]\longrightarrow X$ such that $\gamma\bigl([0,1]\bigr)\subset Y$, that $\gamma(0)=x$ and that $\gamma(1)=y$.