She was supervised by Dr Simo Mthethwa.

‘Initially, I was drawn to UKZN because it was close to home and after recovering from a period of deep depression I preferred to stay near my support system,’ said Nogwebela. ‘The most significant factor, however, was my desire to work with Dr Simo Mthethwa. Having already decided to focus on compactifications, I believed he had the most expertise in this field in South Africa.’ UKZN’s fee remission policy for PhD studies also played a part.

‘Gugulethu’s PhD work uses algebraic machinery to tackle problems arising in classical *topology, and compactifications in particular,’ said Mthethwa. ‘**Compactness is one of the most crucial properties in topology. The nature of the points that are added to the space, the so-called remainder of the compactification, to make it compact, is interesting.’

‘Gugulethu studied compactifications with a zero-dimensional remainder in a constructive manner. The novelty of the results in her thesis lies in the constructiveness of the proofs of results, ie the fact that their proofs mostly do not depend on any choice principle.’

Nogwebela described her research: ‘I dealt with compactifications of frames with zero-dimensional remainder. In classical topology, compactification is a process of making a space compact by adding enough points to the space; we add the points in such a way that if we take the closure of the space being compactified, we get the compactification (this restriction ensures that we do not add too many points). Hence the remainder is the points we are adding to our space to make it compact. We want these points we are adding to our space to be zero-dimensional; that would imply that that the points are totally disconnected, meaning that any two points can be separated by disjoint open sets. I first studied this process in classical topology and then later in point-free topology.’

Nogwebela was introduced to the subject of topology during her honours year by Professor Sheng Bau, who made it both interesting and stimulating. ‘In topology, shapes and curves are considered as collections or sets of points, and I found this perspective more appealing than any other I had encountered,’ she said. ‘I became particularly fascinated with the concept of compactness. When Dr Mthethwa introduced the idea of studying how to make a space compact, I was thrilled. Since then, I have been focusing on compactification, first in spaces and later in frames.’

She explained why her research is significant: ‘Compactness is a property of sets in topology. A subset K of a topological space is called compact if every cover of K has a finite subcover. To visualise this hypothetically, imagine a room painted with infinitely many colors. Compactness means you could find a finite number of colors that would still allow you to cover the room completely.

‘Compact sets behave somewhat like finite or bounded sets in that many techniques applicable to finite sets also work for compact sets. This property has significant implications in various mathematical contexts. For example, a well-known compact set is the closed and bounded interval on the real number line, as described by the Heine-Borel Theorem.

‘One important application of compactness is the Extreme Value Theorem, which states that every continuous function defined on a compact set attains both its maximum and minimum values.’

With her PhD under her belt, Nogwebela has recently taken up a lecturing position at North-West University. ‘I aim to grow within my department and become a better lecturer,’ she said. ‘My long-term goal is to establish myself as a well-regarded researcher and eventually become a professor. I aspire to publish papers as frequently as Professor Themba Dube and to be the kind of supportive supervisor for students that Dr Mthethwa was for us.

‘Completing my PhD in two years required many sacrifices and meant neglecting my personal life. Therefore, I now want to focus more on my personal life and hopefully build a family.’

Nogwebela said her mother, brothers and Mthethwa were crucial to her PhD success. ‘My mother encouraged breaks and reminded me of the long-term benefits of my hard work. My brothers helped me avoid burnout by emphasising the need for rest. Dr Mthethwa’s guidance and stimulating conversations about compactification were also key to my motivation and idea generation.

‘And let me not forget to mention a fellow PhD student, Mr Siyabonga Dubazana, who always assisted with my proofreading even though he had a busy schedule. He also assisted in my thinking process when I was stuck on a problem.

‘I want to see more women pursue postgraduate studies in pure mathematics, and I hope my story can inspire them,’ said Nogwebela.

*Topology: is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

**Compactness:  is a property that seeks to generalise the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no “punctures” or “missing endpoints”, ie it includes all limiting values of points.

Words: Sally Frost

Photograph: Sethu Dlamini