Suppose the object of the category are metric spaces and for $\left(A,d_A\right)$ and $\left(B,d_B\right)$ metric spaces over sets A and B, a morphisms of two metric space is given by a function between the underlying sets, such that $f$ presere the metric structure: $\forall x,y,z \in A$ we have:
- $$ d_B\left(f\left(x\right),f\left(y\right)\right)= 0 \Leftrightarrow f\left(x\right)=f\left(y \right)$$
- $$d_B\left(f\left(x\right),f\left(y\right)\right)=d_y\left(f\left(y\right),f\left(x\right)\right)$$
- $$d_B\left(f\left(x\right),f\left(y\right)\right) \le d_B\left(f\left(x\right),f\left(z\right)\right) + d_B\left(f\left(z\right),f\left(y\right)\right) $$ and furthermore : $\forall \epsilon > 0$, $\exists \delta >0 $ which satisfy:
- $$d_A\left(x,y\right)<\delta \Rightarrow d_B \left(f\left(x\right),f\left(y\right)\right)< \epsilon$$
Is this the category of metric spaces and continues functions? What if we drop the last requirement?