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We define the Möbius band $M$ as the space $I \times I$ when identifying $(0,t)\sim (1,1-t)$ through $\rho: I \times I \to M$. We wish to prove $S=\rho(I \times \{ \frac{1}{2}\})$ is a circle. The solution was to define a function $f:I \times \left\{\frac12\right\} \to S^1$ by $f\left(t,\frac12\right)=(\cos(2\pi t), \sin(2 \pi t))$ and note it is continuous and closed (since it's from a $T_2$ space to a Compact space). How does it help? I assume we try to get a homeomorphism $S \to S^1$ somehow.

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    $\begingroup$ Continuous, closed and bijective means it's a homeomorphism. Do you see why it's bijective? $\endgroup$ – Ayman Hourieh Jan 20 '14 at 21:10
  • $\begingroup$ $f(0)=f(1)$, hence not 1-1. How do we move from talking about $f$ and $I×\frac{1}{2}$ to the space really want, S? $\endgroup$ – Emolga Jan 20 '14 at 21:13
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    $\begingroup$ Use the universal property of quotient maps. $\endgroup$ – Ayman Hourieh Jan 20 '14 at 21:16
  • $\begingroup$ I don't get it. The arrows don't match, and why do we only need to prove the new map is continuous? $\endgroup$ – Emolga Jan 20 '14 at 21:53
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    $\begingroup$ By the way. $f$ is closed because it is a map from a compact to a T$_2$ space, not the other way. $\endgroup$ – Stefan Hamcke Jan 20 '14 at 22:00
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Use the universal property of quotient maps. Please have a look at the following diagram: enter image description here

Here $p$ is the quotient map to the Möbius band and $q$ is the restriction of $p$ on the middle line. Now there are two ways to topologize the set $p\left(I\times\left\{\frac12\right\}\right)$: We can regard it as a subspace of $M$, or we give it the final topology with respect to $q$ which means we see at as a quotient of $I\times\left\{\frac12\right\}$. There are cases where these topologies are different, and we see that they are equal iff $q$ is a quotient map. There are criteria which enable us to decide if this is the case. For example, if $A$ is a closed and saturated subspace of $X$, then the restriction of a quotient map to $A$ is again a quotient map. As $I\times\left\{\frac12\right\}$ is indeed closed in $I×I$ and saturated with respect to $p$, we conclude that $q$ is a quotient map.
Now $f$ as you defined it was continuous, surjective, and was compatible with the relation (meaning that $f(x)=f(y)$ whenever $q(x)=q(y)$). By the universal property of quotient maps it thus factors as $\tilde f\circ q$, where $\tilde f$ is uniquely determined and surjective.
Recall what we need for $\tilde f$ to be injective and open.

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