## anonymous one year ago please can someone tell me what an open and close set is in a topology space ,,,,,,,

1. anonymous

I have no idea sorry :/

2. anonymous

https://en.wikipedia.org/wiki/Open_set Try this website.

3. zzr0ck3r

That is exactly what a topology is. You declare what sets are open

4. zzr0ck3r

Can you be a little more specific in your question?

5. zzr0ck3r

There are many ways to declare sets open, and once you do the topology can be generated. A set is closed if and only if its compliment is open. This stuff can be very confusing at first but just tag me if you want to talk about it. And watch the following video to prepare yourself for the adventure you are about to embark on. https://www.youtube.com/watch?v=SyD4p8_y8Kw

6. anonymous

if you have a set $$X$$ then you usually *define* what sets are open, since openness refers to being in the topology $$\tau$$ you endow the set with. you cannot of course make any random subset of $$P(X)$$ a topology -- it must satisfy some axioms, namely closure under finitary intersections, closure under union, and inclusion of the empty set and the whole set

7. anonymous

so this question is very vague -- in a topological space, the open sets are what we define them to be. closed sets, on the other hand, are simply those whose complement in $$X$$ is open

8. anonymous

now, there are also equivalent ways of defining topological spaces, namely in terms of neighborhoods, closed sets, and also the Kuratowski axioms (in which everything flows from a suitable definition of the 'closure' of a set)

9. anonymous

the idea of an open set in topology, however, has to do with how roughly you can make out or distinguish features of the space

10. anonymous

a super fine topology, made by endowing $$X$$ with $$\tau=P(X)$$ i.e. the power set of $$X$$, is also the topology on $$X$$ *generated* by treating every point $$x\in X$$ as very 'distinguishable'; this 'very fine' topology is called the *discrete topology*

11. anonymous

on the opposite end of the spectrum, we can't distinguish any $$x$$ from each other at the topological level, and the topology is $$\tau=\{\emptyset, X\}$$ -- we can either see nothing at all or everything together, with no real resolving power. this extremely coarse topology is known as the *trivial topology*

12. anonymous

in the standard topology on $$\mathbb{R}^n$$, that which arises naturally from the metric $$d(x,y)=|x-y|$$ (equipping $$\mathbb{R^n}$$ with a metric $$d$$ yields a metric space, which is a topological space with the corresponding metric topology), is generated by open balls $$B_r(x_0)=\{x\in\mathbb{R}^n:|x-x_0|<r\}$$ with all possible centers $$x_0\in\mathbb{R}$$ with positive real radii $$\epsilon>0$$

13. zzr0ck3r

There is no need for a metric as there are uncountable many topological spaces with no metric. There are many ways to talk about open sets in a topological space. You may define them to be open but you must include that arbitrary unions and finite intersections are also open, and you must include the empty set and the entire set. If given a basis or subbasis for a topology then the topology can be generated. A topology is just a collection of open sets by definition. You may define them, or they may be defined. But as long as it passes those conditions it is a topology, and every element of the topology is an open set(yes I mean element, as it is a collection of sets). Quickly you realize that every thing in math is a set :) This question you are asking is a very hard one, and it is what makes topology confusing. Sets are open if you say they are.... I would suggest getting Munkresbook Topology.

14. zzr0ck3r

Here are two of the most basic topological spaces on some set $$A$$. Then we have indiscreet topology $$\mathit{T}$$ = $$\{\emptyset, A\}$$ Obviously the arbitrary unions (infinite or finite) of sets in $$T$$ are in $$T$$ and same goes for finite intersections. And the entire set and the empty set are in $$T$$ and so $$T$$ is a topology. And the discrete topology $$T=\{U\mid U \subset A\}$$ Can you tell me why this is a topology?

15. zzr0ck3r

lol @oldrin.bataku I only saw your last response(stupid phone). So I guess much of this is repetitive but said in a different way...

16. anonymous

thanks all but every set in a topological space is open

17. anonymous

uh, no they're not. every set in a topology is open, but not every set in a topological space is open

18. anonymous

R with the standard topology is clearly a topology space, but [0,1] is closed

19. anonymous

and not open

20. zzr0ck3r

I would say every element of a topology is an open set.

21. anonymous

ok but can you guys help me on matic space and i am so lost in understanding euclidean space

22. anonymous

ask a specific question then? a metric space is a set X with a map d : X -> R that satisfies a few properties 1) d(x,y) >= 0 for all x, y in X 2) d(x,y) = 0 iff x = y for all x,y in X 3) d(x,y) <= d(x,z) + d(y,z) for x,y,z in X 4) d(x,y) = d(y,x) for all x,y in X

23. anonymous

ok, got that but just that i am offering a course mth301 with the link http://www.nou.edu.ng/uploads/NOUN_OCL/pdf/edited_pdf3/MTH%20301.pdf .please rake a look at the course material and see if you can make me understand euclidean space on page 8, page 14 remark, page 16,17,18,19,22

24. anonymous

25. anonymous

it would be helpful if you show examples , or can you make a video on those please?

26. anonymous

Euclidean space is just the standard n-dimensional space you get when you take a product of $$\mathbb R$$ $$n$$-times and as a metric space it uses the standard one derived from the Pythagorean theorem: $$d(x,y)=\sqrt{\sum_{j=1}^n (x_j-y_j)^2}$$

27. anonymous

am so grateful for meeting you here sir

28. anonymous

it uses the standard metric*; the remark on page 14 concerns the fact that a metric space is not just a simple set, it's a set equipped with a notion of distance between its members. this is important because there are ways to turn $$\mathbb{R}^n$$ into a metric space using *other* metrics, and this corresponds to other, more exotic geometries. a perfect example is the one given on page 14, $$d(x,y)=\sum_{j=1}^n |x_j-y_j|$$, which yields a metric space that corresponds to *taxicab geometry*. this is because it measures distances between points in the sense of street blocks, much like a taxicab driver does in a busy city like Manhattan.

29. anonymous

so you see that $$\mathbb{R}^n$$, the space of n-tuples of real numbers, or equivalently $$\{(x_1,\dots,x_n):x_1,\dots,x_n\in\mathbb{R}\}$$, can be given at least two distinct notions of 'distance' between these tuples, one according to the n-dimensional Euclidean geometry (think Pythagorean theorem) and one according to the taxicab geometry

30. anonymous

i know about the Pythagorean theorem but i do not know about taxi cab geometry. can you explain what one should know about Euclidean space?

31. anonymous

and i feel that book does not give a basic introduction for novice like me .... please do you have a better book . am not good at maths but i just love to study it .plz help

32. zzr0ck3r

The study of metric spaces would be better suited for real analysis and I would look into the book Principles of mathematical analysis by Rudin(also known as "baby rudin"). For topology I would suggest Munkres, but he does not get heavy into metric spaces. As far as the taxi cab metric, think of the plane in two dimensions with the regular metric (absolute value). We say the distance between two points is the "strait line distance" between them, and this is why we say the quickest way to get from point A to point B is a straight line. So the distance between (0,0) and (3,4) is 5. But with the taxi cab metric we would say the distance between (0,0) and (3,4) is 7, 3 to the right, and 4 up because if we were in a taxi, we cant drive threw buildings so we have to stay on the streets.

33. zzr0ck3r

Euclidean space is the standard metric. Also note that on the set R (not R^2, R^3....) these two metrics are the same, so it is best to think about this in at least two dimensions.

34. anonymous

ok sir, got the help. have downloaded baby rudin

Find more explanations on OpenStudy