Angel "Java" Lopez en Blog

Publicado el 24 de Mayo, 2016, 5:40

Investigando algunos temas, esta semana me topé con el libro "A Course in Point Set Topology" de John B Conway, matemático americano, dedicado al análisis funcional. Tengo que estudiar ese tema, porque cada vez más va a aparecer en algunos posts que estoy escribiendo, como Matemáticas y Física Cuántica. Es interesante compartir por qué Conway escribe un libro sobre Topología General. He aquí la explicación, al comienzo del prefacio:

Point set topology was my first love in mathematics. I took the course as an undergraduate at Loyola University in New Orleans and my professor, Harry Fledderman, told me to go to the library and solve all the problems in the book while he tutored the other student who had signed up for the course. (Yes, I know it sounds strange today, but there were only two students in the course.) I kept a notebook with my solutions, and once a week I reported for his inspection of my work. I felt like a real mathematician learning real mathematics. It had a great influence on me and made me realize how much I wanted to be a mathematician. Even now I can"t tell you whether the love I have for point set topology was the cause of this feeling or whether that love was a consequence of this learning style. I was disappointed to later discover that research in this area had mostly petered out. I found equally attractive research areas in which to sow my oats, but I always retained this youthful love affair.

Es una forma muy interesante de estudiar matemáticas, y ya me he encontrado más de una vez con algún profesor que adopta este camino para un estudiante brillante.

Más adelante Conwayexplica las elecciones de contenido de este libro. Me gusta como plantea los temas, de lo particular a lo general:

Following my philosophy of beginning with the particular, I start with metric spaces. I believe that these are far easier to connect with students"experience. They also seem to me to be the more prevalent topological spaces used in other areas and are therefore worth extra emphasis. Chapter 2 defines and develops abstract topological spaces, with metric spaces as the source of inspiration. I narrow the discussion by quickly restricting the focus to Hausdorff spaces. Needless to say, some of the more elementary arguments in topological spaces are the same as those in metric spaces. There is no problem here; I just refer students to the metric space proof and invite them to carry out the analogous argument, which in most cases is almost identical.

Y toma una curiosa decisión en el último capítulo: concentranrse en las aplicaciones continuas antes que en las propiedades del espacio en estudio:

Chapter 3 concentrates on continuous real-valued functions. My belief is that the continuous functions on a space are more important than the underlying space. Maybe that"s because I"m an analyst. I know that much of modern topology concentrates on the underlying geometry of a space, but surely that must be saved until after the student has encountered the need.

Tengo que estudiar alguna parte, especialmente Espacios Métricos, para lo que estoy escribiendo.

Nos leemos!

Angel "Java" Lopez
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