I am a Physics undergrad, and just started studying Topology. How do you define neighborhood and open set in Topology. Wikipedia gives a circular definition.
While a neighborhood is defined as follows:. One of the problems of introducing students to topology is that the open set axioms are often taken as THE definition of a topology, when they are quite unintuitive, though extremely useful in the long run. I argue that the neighbourhood definition, while somewhat cumbersome, has the advantage of being closely related to ideas from analysis, and has a historical basis; it is of course as follows:.
The open set definition of continuity is then justified as being equivalent to this definition in terms of neighbourhoods. Students should be aware that there are many approaches to the notion of topology, whose advantages should be compared.
There should be no "take it or leave it" approach, but students should be encouraged to form a judgement, in terms of the character of the theory and its methods. And see which definition is appropriate in which cases. June The above approach is taken in my book Topology and Groupoids , in order to motivate the definition of open set. November 17, Peter Freyd writes in the Introduction to his book Abelian Categories.
The mathematical correctness of such a definition reveals nothing about topology except that its basic axioms can be made quite simple. And with category theory we are confronted with the same pedagogical problem.
A better albeit not perfect description of topology is that it is the study of continuous maps; and category theory is likewise better described as the theory of functors.
I would also like to put in a reference to remarks of Bill Lawvere that the notion of space in mathematics is crucial for the representation of motion. This is illustrated in this lecture Out of line , particularly the section and video on Motion.
To complement the other answers, which tell you what the normal definition of open set in a topology, I'll give another possibility for the definition of neighbourhood in a metric space note that this won't make sense for general topological spaces, but I think it's what's motivating the definition of open set you gave.
Now the definition of open set you've given agrees with the usual one for metric spaces. Usually I'm only thinking about open neighborhoods, but I guess some people find convenient uses for the more flexible definition of a not necessarily open neighborhood.
The definition of a topology in the most abstract point of view is given by the collection of the open sets. It has to verify that. There is also an alternative approach midway between open sets and the neighbourhood systems in Ronnie Brown's answer, one that is quite natural from an order theoretic point of view. The axioms would be much the same as for neighbourhood systems, with an extra infinite union axiom like with open sets, specifically. Note axiom 4 for neighbourhood relations looks a little nicer than the corresponding axiom for neighbourhood systems.
These would satisfy the same axioms except that 5 would only apply to finite collections, i. So neighbourhood relations are really just a slight variant of proximity relations. Home Questions Tags Users Unanswered. Definition of neighborhood and open set in topology Ask Question.
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Active 25 days ago. Viewed 37k times. An open set is defined as follows. How do you define it exactly? There's no single fixed definition of an open set. Of course, commonly used sets like the complex numbers for example have standard topologies, with much tighter definitions of what constitutes an open set. But I need a more mathematical text which does not go into depth. I have been told Armstrong is good.
What do you think? How about Munkres.? But it does have a certain level of rigor. For more information, see Topology by Munkres.
Semi open set in topology pdf
Potato Potato It is possible for a set to be both open and closed. They are all open just because I say so! THEN one can develop the open set axioms and show that one can recover the neighbourhoods.
Peter Freyd writes in the Introduction to his book Abelian Categories "If topology were publicly defined as the study of families of sets closed under finite intersection and infinite unions a serious disservice would be perpetrated on embryonic students of topology. Ronnie Brown Ronnie Brown 13k 1 1 gold badge 31 31 silver badges 43 43 bronze badges.
Open and Closed set in Topology
See math. I am pleased to say my book includes the crucial words "a non-empty set". And thank you so much for making your book available online!
Making accessible both versions of the book was a kind of moral duty in view of the way writing the book and so trying to clarify my understanding as I drafted versions led me into so many research paths. Rudy the Reindeer Rudy the Reindeer It's a very nice book and very easy to read. Glougloubarbaki Glougloubarbaki 6, 1 1 gold badge 15 15 silver badges 40 40 bronze badges.
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