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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 $X$?

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$\pi_0$ is the set of path-components of $X$, and of course you can give it the quotient topology. But for a very large class of spaces path components are open and so the quotient topolgy $\pi_0$ will be discrete.

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    @palio: By definition, every point in a path component has an open neighborhood which is path connected, so that means this neighborhood is contained in the path component. So the path component is open.2012-02-23
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Actually you are counting the path connected components of your topological space, what is important is the cardinality of this set rather than its topology.