[Anindya Bhattacharyya](https://forum.azimuthproject.org/discussion/comment/18556/#Comment_18556)

> I can see how dropping axiom (d) just means the space has indistinguishable points (ie it no longer has to be T0), and dropping (f) just means it no longer has to be connected... but dropping axiom (e) seems to have stranger effects, ones that can't obviously be captured by our usual concept of topology.

Well, in the very least we can think of Lawvere metric spaces as inducing [Bitopological spaces](https://en.wikipedia.org/wiki/Bitopological_space).

Ordinarily, a metric space topology is defined using a metric \$$(X,d)\$$ by taking the set of [*open balls*](https://en.wikibooks.org/wiki/Topology/Metric_Spaces#Open_Ball) as a [topological basis](http://mathworld.wolfram.com/TopologicalBasis.html):

$$B_r(p) = \\{ x \in X \mid d(x,p) < r \\} \text{ where } r \in \mathbb{R}^+$$

This would be the end of it, except in a Lawvere space \$$d(x,p) \neq d(p,x)\$$

So we can construct *another* topology by flipping the usual definition:

$$A_r(p) = \\{ x \in X \mid d(p,x) < r \\} \text{ where } r \in \mathbb{R}^+$$

**Puzzle MD1.** Show these each form a topological basis. That is, show:

1. For each \$$x\$$ in \$$X\$$, there is at least one basis element \$$E\$$ containing \$$x\$$.

2. If \$$x\$$ belongs to the intersection of two basis elements \$$E_1\$$ and \$$E_2\$$, then there is a basis element \$$E_3\$$ containing \$$x\$$ such that \$$E_3\$$ subset \$$E_1\$$ intersection \$$E_2\$$.

**Puzzle MD2.** Morphisms over enriched categories are maps \$$\phi : \mathcal{X} \to \mathcal{Y} \$$ obeying:

\$$\mathcal{X}(a,b) \leq_{\mathcal{X}} \mathcal{X}(c,d) \implies \mathcal{Y}(\phi(a),\phi(b)) \leq_{\mathcal{Y}} \mathcal{Y}(\phi(c),\phi(d)) \$$

\$$\mathcal{X}(a,b) \otimes_{\mathcal{X}} \mathcal{X}(c,d) \leq_{\mathcal{X}} \mathcal{X}(e,f) \implies \mathcal{Y}(\phi(a),\phi(b)) \otimes_{\mathcal{Y}} \mathcal{Y}(\phi(c),\phi(d)) \leq_{\mathcal{Y}} \mathcal{Y}(\phi(e),\phi(f)) \$$

\$$\mathcal{X}(a,b) = I \implies \mathcal{Y}(\phi(a),\phi(b)) = I \$$

Let \$$\mathcal{X}\$$ and \$$\mathcal{Y}\$$ be **Cost**-enriched categories, and let \$$\phi : \mathcal{X} \to \mathcal{Y}\$$ be a morphism between them. Does \$$\phi\$$ induce a [pairwise continuous map](https://en.wikipedia.org/wiki/Bitopological_space#Continuity) over their bitopologies? Or the other way around - does a pairwise continuous map between the bitopologies of \$$\mathcal{X}\$$ and \$$\mathcal{Y}\$$ induce an enriched category morphism?

-------------------

I don't know the answer to **MD2**...