# Origin of the term 'Trace Topology'

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.

• I can't say where it comes from, but we have the analogous construction for other families of sets. If $\mathscr{F}$ is a filter on $X$ with $F\cap A \neq \varnothing$ for all $F\in \mathscr{F}$, then the family $\{ F \cap A : F \in \mathscr{F}\}$ is the trace filter of $\mathscr{F}$ on $A$. If $\mathscr{M}$ is a $\sigma$-algebra on $X$, then $\{ M \cap A : M \in \mathscr{M}\}$ is the trace $\sigma$-algebra of $\mathscr{M}$ on $A$. – Daniel Fischer Nov 21 '16 at 14:59
• @DanielFischer This seems very promising, just wanted to say I read it and I will think and discuss about it somewhat. Thanks – Patrick Abraham Nov 21 '16 at 16:39

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).

• It's a really interesting reading on the word 'topology', but the question was focused on the word 'trace topology' not on 'topology' itself. But since I learned a bunch of interesting facts about topology itself, still upvoted :) – Patrick Abraham Nov 21 '16 at 14:44
• Thanks and sorry for having misinterpreted your question. – Jean Marie Nov 21 '16 at 14:46
• You don't have to be sorry at all, was very informative. Have a good day and best of luck – Patrick Abraham Nov 21 '16 at 14:52