To start studying topology, what basic knowledge should I have?
No prior knowledge is needed. But some mathematical maturity would be very helpful.
You should be comfortable manipulating sets, and you should have mathematical maturity.
For the rest there are not really prequisites, but I highly recommend to learn about metric space analysis first, which introduces concepts like open sets and closed sets etc.
Topology is a generalisation of this, so analysis is important for the motivation of the topology concepts.
Completely true and I basically agree with what you said. But there is a dark side of this moon. The example of metric spaces in mind, usually a Euclidean plane or sometimes a volume, can also be a burden learning topology, simply because metric spaces tend to be the "nice" topologies and there are a lot more which do not have those properties. I found it always a bit challenging to imagine spaces which e.g. aren't Hausdorff. So as basic as metric spaces are, and yes, they are indeed the origin of general topology, as obstructive they can be.
Thus there is no formal requirement to learn metric spaces first. It could as well be a good idea to start without them - without the prejudices they carry along. So the basic concept of sets and functions is strictly all one needs to learn topology.
my first analysis teacher george mackey told a story once, when this discussion came up, of how general the context should be for learning topology. he said a study showed there was no agreement on this point except for the fact that everyone agreed that one should learn the subject in whatever generality the respondent himself had learned it in grad school. so in keeping with this, i recommend learning metric space topology first, then learning general topology, say as in kelley. but i want to emphasize that as i learned at my own pain, one should study specific examples of non trivial topological spaces, such as spheres, lens spaces, compact surfaces not necessarily orientable, grassmannians, SO(3), SU(2), and more general examples constructed by "adding cells", i.e. cell complexes. to repeat, do not be satisfied at memorizing axioms for general topology without doing the harder work of mastering some actual examples.
What topology are you interested in and what is your level?
now i am study in Bachelor of Science (Physics)
I want to apply knowledge about topology to physics problem
I just want to have some knowledge about topology
Can you give an example?
Yes, but this is the question. Basics are found in any "Introduction to Topology" book and will mainly deal with the set theoretic part of it. If you mention physics, then manifolds and their topologies come to mind, which is a very different approach, but very likely presumes the set theoretic basics as given. If you say physics, but mean cosmology, then algebraic topology might be important, which is again another approach. Compared to its age, topology well-nigh exploded into various branches which are each by themselves more or less sophisticated. As a starting point, any book with "Introduction" in the title will probably be a good choice, as these subjects (separation axioms, limit points, compactness etc.) are of general interest and basic to many other realms of topology as well as analysis.
I do not have an example.
but I have an interest in Relativity theory.
I need knowledge of topology before studying relativity theory.
Maybe because general relativity uses topology?
Actually, most areas of general relativity do not make explicit use of topology, and those areas of general relativity that do make explicit use of topology, e.g., global methods (**), typically are not seen during a first go at general relativity.
My recommendation to physics students interested in general relativity: first study linear algebra and multivariable calculus, and then study one of the good introductions to general relativity, e.g., "Gravity: An Introduction to Einstein's General Relativity" by Hartle.
For math students interested in learning general relativity, I might make different recommendations.
Similarly, I do not recommend that, before studying quantum mechanics, physics students learn functional analysis up to and including the spectral theorem for unbounded self-adjoint operators on Hilbert spaces, even though the canonical commutations relations ensure that unbounded operators are used in quantum mechanics.
Again, for advanced math students interested in quantum mechanics, I make different recommendations,
(**) A cute little example uses the open cover definition of compactness to show that all compact spacetimes contained time loops.
I agree with George Jones that at least at first, a formal course in Topology is not needed for much of Physics. If you know multivariable calculus then you know what a limit is and what continuity means. That is enough Topology to get started. Also you must know a little Linear Algebra from the multivariate Chain Rule and the Change of Variables formula for integration.
That said, given multivariate calculus, you could start to learn Differential Geometry for Relativity theory. It seems though that a Physics course on General Relativity would teach the geometry as it goes along so a separate math course might be a distraction albeit an interesting one. Leonard Susskind has Youtube Lectures on General Relativity that develop the differential geometry from scratch.
For Quantum Mechanics the same is true. You can learn the math in a Physics course and the math is much easier than in Relativity.
A Physicist needs to know about differential equations and there are many topological ideas that arise in solving them. For Ordinary Differential equations I started with an old book by Witold Hurewicz called - I think - Lecture Notes on Ordinary Differential Equations. It gives a rigorous foundation and if you wanted to pursue the subject further you might follow up with a course in Dynamical Systems.
For Partial Differential Equations the possibilities are enormous. A first mathematical step that uses topology might be to study the Laplace equation in regions of the plane through Complex Analysis. In this subject point set topology is introduced in the context of mapping properties of analytic functions. Integration theory leads to the concept of homology which is a branch of algebraic topology. The idea of Riemann surfaces introduces manifolds as the natural domains of multivalued complex functions. Meromorphic functions introduce the the study of singularities of a field That's a lot of topology. Complex analysis also has many geometric ideas which are useful in Physics for instance the idea of conformal mapping.
If you want to learn some topology for fun, try the book The Shape of Space by Jefferey Weeks. It introduces the topology of manifolds without technicalities. Anyone can read it and it is simply gorgeous.
thank you everyone
thank you everyone. Now i know Linear Algebra, DEs(ODE and PDE), Calculus I, Calculus II, and CaIculus III.
but i don't know Complex analysis, Differential Geometry.
the basic definition in topology is what is a continuous function? then the larger question is: what does continuity imply?
maybe too elementary but i recommend hilbert's geometry and the imagination. there is a chapter on topology but everything in this book will give you insight into mathematics from a great great mathematician.
The typical text on topology, applied or not, would include much that would not be relevant to your interests. In fact, you might find almost nothing relevant to relativity. You would probably be better off starting your studies of relativity and filling in topology on an as-needed basis.
here is a lecture by the author of lavinia's recommended book, the lecture starts about 6 1/2 minutes into the video.
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