Is a projection a quotient map?(adsbygoogle = window.adsbygoogle || []).push({});

I think a quotient map is an onto map p:X-->Y (where X and Y are topological spaces) such that

U is open/closed in Y iff (p)-1(U) is open/closed in X.

And a projection is a map f:X-->X/~ defined by f(x)=[x] where [x] is the equivalent class (for a relation ~) containing x.

I guess a projection is onto because for every equivalent class [x], there is an x that maps to it, but I am wondering if every projection is a quotient map (or if every projection is continuous).

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# Projection and quotient map

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