Prove that $f^{-1}(N\times N-\triangle)$ is a union of open ball in M.

Question

Let $f:M\to N\times N$ continuous with $M,N$ metric spaces and $\triangle\subset N\times N$ a diagonal. Prove that $f^{-1}(N\times N-\Delta)$ is a union of open ball in M.

If f is continuous, then for all open set $V\subset N\times N$, $f^{−1}(V)$ is open in M. So, I should proof that , $N\times N-\Delta$ is an open set. If I take $(x,y)\notin N\times N-\triangle$ this mean, $x\neq y$, and define $r=\dfrac{d(x,y)}{2}>0$, then $(x,y)\in B((x,y),r)\subset N\times N-\triangle$, but this open ball is up or down the diagonal $\Delta$ (I see this replacing N by $\mathbb{R}$). I'm not sure if this means $N\times N-\Delta$ is open, because the open ball isn't entirely contained .

Related: $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed — Martin Sleziak, May 08 '16 at 02:06

score 1 · Answer 1 · answered Sep 16 '15 at 02:46

1

Correction: $f$ is continuous, then for all open set $V\subset N\times N$, $f^{-1}(V)$ is open in $M$. Thus we can choose $U=f^{-1}(V)$ in your statement.

Hint: You want to show that $N\times N-\Delta$ is open in $N$. From there, $f^{-1}(N\times N-\Delta)$ is open in $M$ (so is union of open balls).

answered Sep 16 '15 at 02:46

Quang Hoang

16,174

Sorry, for the wrong definition. And, my problem is how I can determine if $N\times N-\triangle$ is a open set??. Thanks! – Ahna Akbar Sep 16 '15 at 02:52
Take a pair $(x,y)\not \in N\times N-\Delta$, that is $x\ne y$. Can you find an open ball at $(x,y)$ that lies entirely in $N\times N-\Delta$? – Quang Hoang Sep 16 '15 at 02:54
Intuitively I think no, because I will take $r=min(x,y)$, and then the open ball $B((x,y); r)$ meets $\triangle$. – Ahna Akbar Sep 16 '15 at 03:18
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Oh, but I can define $r=\dfrac{d(x,y)}{2}$, and then $B((x,y);r)\subset N\times N-\triangle$. – Ahna Akbar Sep 16 '15 at 03:23
And this implies that $N\times N-\triangle$ is an open set. I'm right? – Ahna Akbar Sep 16 '15 at 04:05
Yes, that's right – Quang Hoang Sep 16 '15 at 04:06
Sorry, one last question, is sufficient to say that the ball $B((x,y),r)\cap\triangle=\emptyset$, with $r=\dfrac{d(x,y)}{2}$??. And thanks for the patience! – Ahna Akbar Sep 16 '15 at 04:21
If exists $z'\in B(z,r)\cap\triangle$, then $z'\in B(z,r)$ and $z'\in \triangle$, in particular if $z'\in B(z,r)$, then $d(z,z')<\dfrac{d(z,z')}{2}$ contradiction.. – Ahna Akbar Sep 16 '15 at 04:34
I think you should review the distance on $N\times N$ or use @MarkJoshi answer above. – Quang Hoang Sep 16 '15 at 04:37
Really?, then what is the distance I should use?? – Ahna Akbar Sep 16 '15 at 04:46

score 1 · Answer 2 · answered Sep 16 '15 at 03:15

1

A metric is continuous in its own topology. So $d^{-1}(0)$ is closed so $\Delta$ is closed.

So its complement is open and so the inverse image of the complement under any continuous function is open.

answered Sep 16 '15 at 03:15

Mark Joshi

5,724

Prove that $f^{-1}(N\times N-\triangle)$ is a union of open ball in M.

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