Frederic Schuller on Metric and Topological Spaces

1:16:57 On metrics that don't measure and hair-dressers! See Intransitive dice.

See Tensor Fields and Simplicial Complexes and observable in nlab:

In physics and in the theory of dynamical systems (deterministic, stochastic, quantum, autonomous, non-autonomous, open, closed, discrete, continuous, with finite or infinite number of degrees of freedom…), an observable is a quantity in some theoretical framework whose value can be measured and observed in principle. Any good theoretical framework of physical phenomena should come with carefully established notion of an observable. ...

See also measurement in nlab:

In principle, any interaction between two physical systems is a measurement, whereby each system measures the state of the other. In practice, we break the symmetry by treating one system as the real physical system S of study and the other system as merely a measuring apparatus A with the sole purpose of measuring S. If the properties of A are more directly observable than those of S, then this is useful. ...

Those two entries definitely need some iteration! 

1:30:19 This bit about homeomorphism and isomorphism in sets is interesting.  In set theory without urelements if you were to define an isomorphism as a structure-preserving map then isomorphism would just reduce to set equality, so the next weaker notion of equivalence you can get is equipotence or just that the two sets are isomorphic if there is a bijection between them. In a sense, urelements are dual to proper classes. So could you define cardinality in another way? I had a vague inkling of this, but it's gone now. If you have only proper classes that you can define with some sort of constructive process, then maybe you could avoid the sort of explosion that produces "ultraexacting cardinals": see Large cardinals, structural reflection, and the HOD Conjecture by Juan P. Aguilera, Joan Bagaria and Philipp Lücke.

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