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In the book - Foundations of differentiable manifolds and Lie groups by Frank Warner, the definition of a slice is as under.

Suppose that $(U,\phi)$ is a coordinate system on $M$ (dimension $d$) with coordinate functions $x_1,...,x_d$, and that $c$ is an integer such that $0\leq c\leq d$. Let $a=(a_1,...,a_d)\in\phi(U)$, and let $S=\{q\in U\ :\ x_i(q)=a_i,i=c+1,...d\}$. Then the subspace $S$ of $M$ together with the coordinate system $\{x_j|_S:j=1,...,c\}$ forms a manifold which is a submanifold of $M$ called a slice of the coordinate system $(U,\phi)$.

Now, it seems to me that, even though in the definition we are fixing the last few coordinates, we could do the same to any random coordinates (not necessarily the last few) and still we would get a slice (if we can call that one).

Warner next proves a proposition that : Let $\psi:M^c\longrightarrow N^d$ be an immersion and let $m\in M$. Then there exists a cubic centered coordinate system $(V,\phi)$ about $\psi(m)$ and a neighbourhood $U$ of $m$ such that $\psi|_U$ is 1:1 and $\psi(U)$ is a slice of $(V,\phi)$.

He follows this by a remark in which I have a doubt. The remark is as followsenter image description here

I don't understand this example. Isn't $\psi(M)\cap V$ a union of two slices, the x-axis portion and the y-axis portion? Any help will be appreciated! Thanks in advance.

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As for the first question, since a permutation of the coordinates is a diffeomorphism, any set of coordinate functions is admitted. It's just easier to write it down that way.

As for the second question, note that the $y-axis$ is divided in two parts which meet a the orign (and the origin is excluded from the set, note the tip of the arrows).

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  • $\begingroup$ But the image $\psi(M)$ contains the origin, it is the image of $m$, so isn't $\psi(M)\cap V$ the figure which looks like a big +. I thought $\psi(M)\cap V=\{(x,y)\in V| x=0\}\cup\{(x,y)\in V|y=0\}$, which is a union of two slices right. $\endgroup$ – gradstudent Apr 15 '14 at 8:30
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    $\begingroup$ Yes, $\psi(M)\cap V$ is the figure which looks like a big $+$. And this is something one does not want to have as a slice, you only want to have the $-$ part (which is $\psi(U)\cap V$ for some subset of $U\subset M$. The example is probably not well chosen, because you have in fact a union of (three) slices. The point is to emphasize that in the theorem you can only claim to really have a slice if you restrict the domain. $\endgroup$ – Thomas Apr 15 '14 at 8:43

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