Let $X \subset \mathbb{R}^2$ be a triangle equipped with the topology induced by the euclidean topology on $\mathbb{R}^2$ and let $Y \subset X$ be the subset made of two sides of the triangle. I need to show that $Y$ is not a deformation retract of $X$.
This is an exercise from the first lectures of my course, so I'm not meant to use homotopy groups or that stuff. My tools are not much more than the definitions (and basic topology, of course).
The claim looks intuitive to me, but I'm stuck in trying to formalize it:
It's pretty easy to show that $Y$ is indeed a retract of $X$, so I need to show that, given a retraction $r:X\to Y$ and letting $i: Y \hookrightarrow X$ be the canonical inclusion, we have $ i \circ r \not \sim id_X$, where $id_X: X \to X$ is the identity function on X.
I though I could reason by absurd, assuming there is an homotopy $F:X \times [0,1] \to X$ between $i \circ r$ and $id_X$ and then get a contradiction with the continuity, i.e. showing that there is an open set of $X$ whose counter image through $F$ is not an open set of $X \times [0,1]$. However I'm having problems in finding such a set. Moreover it seems to me that I'm making the problem harder than it is…any help?