Because of this Euclidean feature, very often although unfortunately not always , a differentiable structure can be put on manifolds, and geometry which is the study of local properties can be used as a tool to study their topology which is the study of global properties.
Of course, algebraic tools are still useful for these spaces. The study of 1- and 2-manifolds is arguably complete — as an exercise, you can probably easily list all 1-manifolds without much prior knowledge, and inexplicably, much about manifolds of dimension greater than 4 is known.
However, for a long time, many aspects of 3- and 4-manifolds had evaded study; thus developed the subfield of low-dimensional topology, the study of manifolds of dimension 4 or below. This is an active area of research, and in recent years has been found to be closely related to quantum field theory in physics.
It is great to study topology at Princeton. Princeton has some of the best topologists in the world; Professors David Gabai, Peter Ozsvath and Zoltan Szabo are all well-known mathematicians in their fields. The junior faculty also includes very promising young topologists.
Gabai has been an important figure in low-dimensional topology, and is especially known for his contributions in the study of hyperbolic 3-manifolds. After finishing the sequence MAT and MAT , topology students can consider taking a junior seminar in knot theory or some other topic , or, if that is not available, writing a junior paper under the guidance of one of the professors. Both junior and senior faculty members are probably willing to provide supervision. It is also a good idea to learn Morse theory, which is an extremely beautiful theory that decomposes a manifold into a CW structure by studying smooth functions on that manifold.
The graduate courses are challenging, but not impossible, so interested students are recommended to speak to the respective professors early. It may also be beneficial to learn other related topics well, including basic abstract algebra, Lie theory, algebraic geometry, and, in particular, differential geometry. The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples.
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General topology normally considers local properties of spaces, and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Let's put the issue of zero infimum aside for now. Now what's wrong with this construction? It contradicts my intuition about what space means.
I expect that if I attach something to a space and then ignore the added part, I get the original space back. So the objects in the category would be "sets with many equivalent metrics on them". Translating this condition to the language of one chosen metric leads to the usual epsilon-delta definition of continuous maps.
As soon as you get continuous maps, topological spaces are not far ahead. Thinking in epsilon-delta terms is almost the same as thinking about open sets. The definition of a topological space is more or less the most general thing for which there is a notion of continuity.
The modern definition is surely not the first one, but all previous were quite close. It was not produced while studying metric spaces, but rather when trying to formalize glueing spaces together and quotienting by subspaces. At least that's my interpretation: historically there were a lot of theorems about spaces glued from polyhedrons, and later for CW-complexes, so the convenience of glueing is absolutely necessary for the definition of a space.
It seems nobody mentioned spaces of distributions. These are duals of function spaces and they are endowed with the weak-topology. In general this topology is not metrizable. This is a fundamental construction in the modern theory of PDEs and there are plenty of books with many results Hormander, Gel'fand and Shilov etc.
See also this answer. A lot of physicists claim that the abstraction to topological spaces is uninteresting, since all topological spaces that arise naturally in physics are metrizable homeomorphic to a metric space. The only interesting thing about the space is its topology, and its topological properties note that the same is true in the field of algebraic topology. So it makes no sense to treat it as a metric space. The hypotheses might look a little complicated, but they are so mostly for the sake of generality.
Intuitively, minimization of functionals should already be interesting. Moreover, these types of problems appear when one is looking for the existence of solutions to PDEs. Sometimes it suffices to minimize some sort of "energy" functional on a Sobolev space or some space of distributions , which satisfies conditions such as the ones above, in order to conclude that some problem e. A very well-known application of this sort is Douglas' solution to the Plateau Problem.
I'm not sure if the proof of Gelfand representation could be made significantly easier in this case, though. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams?
Learn more. Why are topological spaces interesting to study? Ask Question. Asked 8 years, 3 months ago. Active 11 months ago. Viewed 5k times. Question 2: History of topological spaces I mentioned that the abstraction from metric spaces to topological spaces does not seem very natural to me. Question 3: Applications of non-metric topology outside topology I mentioned earlier that "we get a lot of uninteresting spaces". Thank you. Stefan Hamcke
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