An example regarding the Hausdorffness of quotient spaces I was reading Topology by K. Jänich. The following example was bugging me. It seems to me that neither of the two examples is Hausdorff. Can anyone explain to me which one is Hausdorff and why it is? Thanks! 

 A: Maybe even Jänich's sketches are misleading. :)
I suppose you think that the left image shows something like vertical lines $V_c=\{\,(c,y)\mid y\in\Bbb R\,\}$ for each $c$ with $|c|\ge 1$, together with wiggly lines $W_c=\{\,(x,x^3+c)\mid -1<x<1\,\}$ for each $c\in \Bbb R$ (whereas on the right we have $W_c'=\{\,(x,c-x^2)\mid -1<x<1\,\}$). Or perhaps we drop $V_{\pm1}$ and use $-1\le x\le 1$ for $W_c$ and $W_c'$ - it doesn't matter for the question at hand: With these interpretations, both quotient spaces are non-Hausdorff. Indeed, any neighbourhood of $W_c$ (or $W_c'$) contains all $V_c$ with $c\approx \pm1$, i.e., $W_0$ and $W_1$ (or $W_0'$ and $W_1'$) cannot be separated.
However, I suggest a different interpretation (and the hint that this is the intended interpretation may be the innocuous mentioning of "closed submanifold" in the text):
Consider $$V_c=\{\,(c,y)\mid y\in\Bbb R\,\}\qquad \text{for }|c|\ge\frac\pi2$$ and for the left image wiggly lines $$W_c=\Bigl\{\,(x,c+\tan x)\Bigm|-\frac\pi2<x<\frac\pi2\,\Bigr\}\qquad\text{for }c\in\Bbb R,$$
whereas for the right image we take
$$W_c'=\Bigl\{\,(x,c-\tan^2 x)\Bigm|-\frac\pi2<x<\frac\pi2\,\Bigr\}\qquad\text{for }c\in\Bbb R.$$
What happens now is that the left quotient space is homeomorphic to $\Bbb R$ (check this!), whereas in the right quotient space we cannot find separating neighbourhoods for $V_{-\frac\pi2}$ and $V_{\frac\pi2}$: The neuughbourhoods will always contain all $W_c'$ for $c$ large enough.
