Is there a searchable database of mathematical objects that you can search by property?

Question

For example, I could search for functions that are continuous, but that don't have differentiability, and come up with a continuous non-differentiable function. Or a smooth but non-analytical function. Or an commutative monoid without inverses. Or a compact topological space that is countable.

In cases where the object has been proven or conjectured to not exist, it could bring up the theorem or conjecture that states so. For example, if I search for differentiable non-continuous functions, it would bring up this theorem perhaps.

Does such a thing (or something close to it) exist?

Answer 1

This database exists and is called the web. Googling

1. Continuous non-differentiable function leads to Weierstrass function:

In mathematics, the Weierstrass function is an example of a pathological real-valued function on the real line. The function has the property of being continuous everywhere but differentiable nowhere.

2. Smooth but non-analytical function leads to Non-analytic smooth function, where you can find several detailed examples.

3. Countable compact topological space leads to Countable compact spaces as ordinals, a beautiful answer on this site.

Theorem 4. Every countable compact Hausdorff space is homeomorphic to some well-ordered set with the order topology.

4. I have to confess that Commutative monoid without inverses does not give a direct link, probably because $0$ is always its own inverse. But $(\mathbb{N}, +, 0)$ is a standard example of a commutative monoid in which no nonzero element has an inverse.