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Let $(Y, \tau)$ be a topological space and let $X$ be a set such that there exist a surjective function $f \colon X \to Y$.

Consider $\tau_1$ the smallest topology in $X$ that makes $f$ a quotient map,

$\tau_2$ the smallest topology $X$ that makes $f$ continuous,

$\tau_3$ the smallest topology in $X$ that makes $f$ an open map,

$\tau_4$ the smallest topology in $X$ that makes $f$ a closed map,

$\tau_5$ the smallest topology in $X$ that makes $f$ an open and closed map,

$\tau_6$ the smallest topology in $X$ that makes $f$ a closed and continuous map.

I am asked to compare each topology, I understand the definition for "the smallest topology" but it still troubles me, because I don't know how to use it. Can someone give me an example of comparing these topologies?

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    Your $\tau_1$ and $\tau_2$ seem to be the same thing; was one of them supposed to be something else?2011-12-12
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    i corrected. $\tau_1$ is for quotient map2011-12-12

3 Answers 3