Paths between tuples, MSV, decision trees
Consider any path from $a \in T$ to $b \in T$ inside $Y$. Let $t$ be the first time in which the first $k$ coordinates are not distinct. Prior to time $t$, for each $1 \leq i \leq k$, all coordinates in $j \in \{k+1,\ldots,n\}$ such that $a_j = a_i$ must keep satisfying this relation along the path, since otherwise there will be more than $k$ distinct values. Therefore at the point $t$, there will be fewer than $k$ distinct values. We conclude that the first $k$ coordinates always remain distinct, and that they determine the rest of the vector. In particular, the values in the first $k$ coordinates can't cross each other: if $a_i &lt; b_i$ and $a_j &gt; b_j$ then the intermediate value theorem shows that at some point $x$ along the path, $x_i = x_j$. We conclude that $a_1,\ldots,a_k = b_1,\ldots,b_k$. Since the first $k$ coordinates determine the rest, in fact $a = b$.