Motivation for the question:
In our current lecture we use the term 'trace topology' instead of 'subspace topology'
Since I tend to try to understand such terminology, origin and applications to understand the big picture a bit more, I was searching for the origin of the term.
And I honestly didn't found a good explanation for it at all.
Every source always directly mentions it as a 'subspace topology' and mostly ignores the term 'trace topology' and if it is used, the origin is never explained, it's always viewed as a 'subspace topology'.
Where does the term 'trace topology' originated from and what role does the term 'trace' fulfil.
The term "topology" in its modern meaning can be traced to the Polish school of mathematics, Kuratowski (1920s), although it had been coined before by the German school (used by Hausdorff in particular).
The French school has been active in the emergence of the domain, around Poincaré, and young mathematicians like Borel, Lebesgue, Baire, Fréchet... They had the deep feeling that there was an "autonomous" domain do be digged at the roots of analysis. Poincaré coined the word "Analysis situ" for it; it has not survived him. Poincaré in his work on the 3 bodies problem has had typically a pre-topological approach.
Around 1900, there was a general quest for foundations in Mathematics (it was also the case in Physics, as we know). Borel had a very neat approach of what has become "measure theory"; it is not by mere chance that Lebesgue built his new theory of integration, based on measure theory, circa 1905.
Remark: Etymologies of "topology" and "analysis situ" are almost the same (Greek origin for the former, Latin origin for the latter): "topos" is a place and "situs" also, "logos" and "analysis" have also the same meaning.
A good reference: (Origins of the modern definition of topology).