Implications.

**Showcase_22** · January 26th 2010, 03:18 PM

Is at least one of the following two implications valid for every map $f:M \rightarrow N$ of a metric space $M$ to a metric space $N$ ?

a). $f$ is continuous $\Rightarrow \ f(U)$ is open for every $U \subset M$ .

b). $f(U)$ is open for every open $U \subset M \Rightarrow \ f$ is continuous.

I think that a). is valid. The definition of continuity, at a point $c \in M$ , adapted to this case is:

$(\forall \varepsilon >0)(\exists \delta >0)[d_M(c,x)< \delta \ \Rightarrow \ d_N(f(c),f(x))< \varepsilon]$

$d_N(f(c),f(x))< \varepsilon$ is just $B_{\varepsilon}(f(c),d_N)$ valid $\forall c,x \in M$ .

Hence it does imply that $f(U)$ is open for every $U \subset M$ .

So, i've answered the question with a yes! However, for the sake of curiosity I will try and figure out b).

Since a). is true, i'll try and find a counterexample.

Consider the French Railway Metric. This is open because a ball can be constructed around every point.

However, $f$ is not continuous.
Suppose that $x=(1,0)$ and $x_n=(1+\frac{1}{n},0)$ .

As $d(x,x_n) \rightarrow 0$ , $d_F(x,x_n) \rightarrow 2$ so it cannot be continuous.

Is this the correct way to think about the problem?

**Defunkt** · January 26th 2010, 05:45 PM

Originally Posted by Showcase_22

I think that a). is valid. The definition of continuity, at a point $c \in M$ , adapted to this case is:

$(\forall \varepsilon >0)(\exists \delta >0)[d_M(c,x)< \delta \ \Rightarrow \ d_N(f(c),f(x))< \varepsilon]$

$d_N(f(c),f(x))< \varepsilon$ is just $B_{\varepsilon}(f(c),d_N)$ valid $\forall c,x \in M$ .

Hence it does imply that $f(U)$ is open for every $U \subset M$ .

How? To prove that $f(U)$ is open, you need to take a point $n \in f(u)$ and show that there exists an $\epsilon_n > 0$ such that $B_{d_N}(n, \epsilon_n) \subseteq f(U)$ . Have you done that?

After you understand why your proof is wrong, think of $M=N=(\mathbb{R}, d_2)$ with $f(x)=x$ .

Consider the French Railway Metric. This is open because a ball can be constructed around every point.

However, $f$ is not continuous.
Suppose that $x=(1,0)$ and $x_n=(1+\frac{1}{n},0)$ .

As $d(x,x_n) \rightarrow 0$ , $d_F(x,x_n) \rightarrow 2$ so it cannot be continuous.

Is this the correct way to think about the problem?

What is open? You haven't specified any space, just the metric!

**Drexel28** · January 26th 2010, 06:12 PM

a) If $f:\left(X,d\right)\mapsto\left(\bar{X},\bar{d}\rig ht)$ is continuous, then it's open.

Take $X=\bar{X}=\mathbb{R}$ and the usual metric and consider $f(x)=0$

If a function is open, it is continuous.

It is relatively easy to prove that if $f$ is bijective and open then $f^{-1}$ is continuous. Thus, to state the above would imply that every bijective function with continuous inverse is continuous. Let $g:[0,1]\cup\left(2,3\right]\mapsto[0,2]$ be such that $x\mapsto\begin{cases} x & \mbox{if} \quad x\in [0,1] \\ x-1 & \mbox{if} \quad x\in (2,3]\end{cases}$ . This is a bijection which is continuous but has a non-continuous inverse. So...

**Showcase_22** · January 26th 2010, 07:56 PM

Take and the usual metric and consider

$f<img src=$ \mathbb{R},d) \rightarrow (\mathbb{R}, \overline{d})" alt="f

\mathbb{R},d) \rightarrow (\mathbb{R}, \overline{d})" />

$f(x)=0$ maps every point to 0. The range is just the set $\{ 0 \}$ which is closed.

$f(x)=0$ is continuous so we have a counterexample to the question!

every bijective function with continuous inverse is continuous.

The function you've written contradicts the above quote!

**Drexel28** · January 26th 2010, 08:01 PM

Originally Posted by Showcase_22

$f<img src=$ \mathbb{R},d) \rightarrow (\mathbb{R}, \overline{d})" alt="f

\mathbb{R},d) \rightarrow (\mathbb{R}, \overline{d})" />

$f(x)=0$ maps every point to 0. The range is just the set $\{ 0 \}$ which is closed.

$f(x)=0$ is continuous so we have a counterexample to the question!

The function you've written contradicts the above quote!

Yes. Both are wrong. I was providing counterexamples to my quotes.

**Showcase_22** · January 26th 2010, 08:15 PM

Thankyou very much!

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