let $X$ be a topological space. we define an equivalence class on $X$ by $x\sim y$ if there exists a path $\gamma:I\to X$ that joins $x$ to $y$. now the zeroth homotopy set is the quotient $\pi_0(X)=X/_\sim$. My question is why we call it a set isn't it a topological space with the quotient topology induced from $X$?
zeroth hompotopy set of a topological space
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algebraic-topology
connectedness