Robert Moore had been appointed as associate professor at the University of Texas in 1920 and it was Moore who taught Whyburn calculus early in his university studies. Moore quickly saw the mathematical potential in Whyburn, and Whyburn was soon attending further mathematics courses given by Moore who encouraged him greatly towards the study of mathematics. Even before he obtained his first degree in chemistry, Whyburn was undertaking research in mathematics with Moore. Of course with Moore having a deep interest in topology, that was the direction that Whyburn took and it was to become the topic of his research throughout his life.
Whyburn had other connections with mathematics at the University of Texas in addition to his work with Moore. His elder brother, William Marvin Whyburn (1901-1972), was also at the University of Texas at the same time as Gordon Whyburn, and unlike Gordon he was studying mathematics. William Whyburn went on to become Chairman of the Mathematics Department at the University of California, Los Angeles and then at the University of North Carolina. His research was mostly on second order ordinary differential equations, see  for details.
Another mathematician at Texas at this time was Lucille Smith, who was also from Lewisville, and Whyburn married Lucille in 1925. Lucille Whyburn has, since her husband's death, written four fascinating papers (see for example  and ) relating to mathematicians and events from her life with Gordon Whyburn. Despite having these connections with mathematics and despite Moore strongly encouraging him to move from chemistry to mathematics, Whyburn continued with his chemistry studies being awarded his Master's Degree in 1926. Only at that point did he see that continuing to study chemistry was foolish when he had completed high quality research in mathematics.
Whyburn presented his first paper, which was on cyclic elements for locally connected plane continua, at the Western Christmas Meeting of the American Mathematical Society in Chicago on 31 December 1926 (see  for more details). The road to a doctorate in mathematics was now easy, and indeed he was awarded his Ph.D. in mathematics in 1927. Following this he was appointed as adjunct professor of mathematics at the University of Texas, holding this post from 1927 until 1929.
In 1929 Whyburn, together with his wife, was able to go to Europe, financed by a Guggenheim Fellowship, where he spent academic year 1929-30. He spent most time in Vienna working with Hahn, but also visited Warsaw and made important links with Kuratowski and Sierpiński. In  letters between Whyburn and Robert Moore are discussed. These concern Moore's arcwise connectivity theorem which he had first proved for connected open sets in 1916 but had still not published by 1930.
Back in the United States, Whyburn was appointed associate professor of mathematics at Johns Hopkins University, then in 1933 he was approached by the University of Virginia who asked him to accept an appointment as professor and chairman of the Department of Mathematics. He accepted the appointment in 1934 and planned :-
... to get together a few young and congenial mathematicians of topflight accomplishments. their fields should be different but overlapping so that the students would not be confronted with choosing between absolutely unrelated areas. The plan worked beautifully. E H McShane joined the department in 1935, and G H Hedlund in 1939. The three of them ran a program of charm and high standards.Whyburn's first research contributions were on cyclic elements and the structure of continua. This work aimed at examining a locally connected plane continuum and the regions of the plane created by it. The theory was based on cyclic elements, that is a region C such that any two points of C are contained in a simple closed curve of C. Around the time Whyburn went to the University of Virginia he began working on homology theory and examined different notions of convergence in the space of all subsets of a compact metric space.
From around 1936 Whyburn looked at open maps and their applications to complex function theory. In 1942 he published his famous text Analytic Topology in which he says :-
Analytic topology is meant to cover those phases of topology which are being developed advantageously by methods in which continuous transformations play the essential role.A reviewer of the book  wrote:-
Were the volume a mere collection of the theory developed in diverse papers in recent years, it would be worthwhile as a source book for present and future workers in transformations. However, it is much more than that. Some of the results are new and much of the treatment is new, some of the proofs acquiring an elegance and polish they sadly lacked in the original papers.The book was the result of further developing material which Whyburn had presented as American Mathematical Society Colloquium lecturer in 1940.
Later major texts by Whyburn were Topological analysis (1958) and Dynamic topology which was jointly authored by Edwin Duda and was published 10 years after Whyburn's death. The term "dynamic topology", writes Whyburn :-
... refers to the body of results of a topological nature in which the function concept plays a central role.Although Whyburn spent his career from 1934 at Virginia, he did make frequent visits for summer teaching. He taught at Stanford, the University of California, UCLA and the University of Colorado. In addition he took study leave in 1952-53 at Stanford, and 1956-57 spent in England and Switzerland. He retired from his role as chairman at Virginia after he suffered a heart attack in 1966. He regained his health but died three years later of a heart attack
Whyburn's character is summed up in :-
Whyburn was a very private man. He was quiet and shy, and remarkably gentle with students and family. But in moments of administrative crisis, he could be extremely tough when he had to be. A man of brilliance, with a remarkable speed in research, he nevertheless believed deeply in continuity and patience, and that it was the total record of accomplishment of a lifetime that mattered most.
Article by: J J O'Connor and E F Robertson