The only biographic information I could find can be found in Les Séminaire de Mathématiques 1933-1939 (Première Partie-l'Histoire) by Michèle Audin:
Ni normalien ni agrégé, Antoine Appert avait passé sa thèse en 1934 [...] et était devenu membre de la SMF la meme année.
Neither a "normalien" nor an "agrégé", Antoine Appert had passed his thesis in 1934 [...] and had become a member of the SMF the same year.
From the same source we learn that he spoke at least twice at the Hadamard seminars (in 1934-1935 and 1936-1937). The list of seminary 'subscribers', i.e. people who were to receive the proceedings, records Appert's presence in the fourth year (1936-1937) and in the fifth year, from which we learn that he lived at 8 rue Berthier in Versailles.
He discussed his thesis Propriétés des espaces abstraits les plus généraux on March 17, 1934, with Cartan as chairman and Fréchet and Valiron as examiners.
From the date of discussion, one can guess the date of birth: Jean Leray and Jean-Louis Destouches discussed their theses in 1933: the former was born in 1906, the latter in 1909; Chevalley discussed it in 1934 and was born in 1909. For Appell, one can conjecture a probable date of birth between 1905 and 1910 [EDIT: I was optimistic in estimating such a narrow range, but I was not far wrong: the correct date seems to be 1903, see below]; certainly not later than 1912 (Weil discussed his thesis at the age of 22, but this is a rather rare case) and reasonably not earlier than 1894 (Raphaël Salem discussed his thesis at the age of 40, but this too is a rather rare case).
Regarding the value of his thesis work, one can read L'entre-deux-guerres mathématique à travers les thèses soutenues en France by Juliette Leloup:
Two doctoral theses deal with questions relating to abstract spaces with reference to Maurice Fréchet. First of all, in 1930, there is the thesis of a Japanese student, Kijiro Kunugui, then in 1934, that of a French student, Antoine Appert, respectively Sur la théorie du nombre de dimensions, 1930 and Propriétés des espaces abstraits les plus généraux, 1934. These two doctorates were based on some of Maurice Fréchet's work, including the latter's 1908 thesis and especially his articles on abstract sets published in the 1920s, the results of which were summarised without demonstration by Fréchet in Les espaces abstraits et leur théorie considérée comme introduction à l'analyse générale in 1928. The two doctoral candidates, from their introduction, explicitly refer to this book which appears as the reference work on the theory of abstract sets. Fréchet also reviews all the publications on the subject. (p. 214)
Moreover, in the thesis report, Maurice Fréchet explains that he himself has already obtained extensions of the properties of linear space to the most general spaces (V) and that he "was able to gather a certain number of them in [his] book 'Les Espaces abstraits'. But for lack of space, [he] could only accommodate the statements. The demonstrations remained scattered in a large number of periodicals". Explaining Appert's role, he writes:
"It is a first merit of M. Appert, to have been able to gather here, in a coherent presentation, statements and demonstrations of properties already known about spaces (V). It could not be a simple compilation. The published demonstrations, due to various authors, overlapped one another and used definitions that were still different at a time when the terminology was evolving. Mr. Appert was able to carry out this work thanks to a well-informed critical sense and a rigour that was always on the alert. But Mr Appert did not stop at writing this paper, which filled a gap. His second merit is to have introduced new notions and obtained interesting properties in the field of which he was taking stock. (pp. 215-216)
Another comment is here.
Some more information about his life can be deduced from the list of his works:
This is interesting because from the title page we learn that Appert was "ancien maîtres de conférences à la Faculté des Sciences de Rennes", i.e., at the time (1951) he was a former lecturer in Rennes.
From these two entries we learn that in 1961-1963 Appert was in Angers
The Editor's note is particularly interesting:
C’est G. Choquet qui a bien voulu nous communiquer le présent article. Il nous écrivait, le 4 avril 1981, à propos de cette Note de A. Appert : "Venant d’un disciple direct de Fréchet, qui a beaucoup réfléchi sur les fondements de la topologie, sa Note est un témoignage sur les préoccupations qui ont été celles des fondateurs de la topologie. A ce titre il m’a semblé qu’elle pourrait intéresser des historiens des sciences et les mathématiciens ou philosophes qui réfléchissent à la naissance des concepts."
This article was kindly communicated by G. Choquet. On April 4, 1981, he wrote to us about this Note by A. Appert: "Coming from a direct disciple of Fréchet, who has given much thought to the foundations of topology, his Note is a testimony to the concerns of the founders of topology. As such, it seemed to me that it could be of interest to historians of science and mathematicians or philosophers who reflect on the birth of concepts."
Appert is also briefly cited in Récoltes et Semailles by Grothendieck:
I had juggled with sets that I called “measurable” (without having ever seen one that was not...) and with convergence nearly everywhere, but I didn’t know what a topological space was. I was a bit lost amonst a dozen equivalent notions of “abstract space” and compactness which I fished out of a little textbook (by someone called Appert, I believe, published in the Actualit´es Scientifiques et Industrielles), which I had come across God knows how. I still had never heard the strange and barbarous words group, field, ring,
module, complex, homology (etc.) pronounced in a mathematical context, and suddenly, they were all rolling over me together. It was quite a shock!
The information above is consistent with what is reported on this site.
In short: Antoine Appert would be the grandson of Félix Antoine Appert (1817-1891), general of the army, French ambassador to the emperor of Russia. He had four children, among them Félix (Etienne Félix Eugène) (1860-1914), colonel, married to Germaine Le Riche de Breuilpont, decorated with the Légion d'honneur in 1902, died for his country in WWI.
Antoine (Antoine Isidore Marie Joseph) Appert, son of Félix, was born August 22, 1903 in Vannes; Docteur es-sciences mathématiques, researcher at C.N.R.S.; married with Marie-Louise Mayaud.
Died on December 16, 1992 in Saint-Laurent-de-la-Plaine, aged 89.