### Topological Spaces and Open Sets

In the previous post, when I talked about continuity, I based this on the notion of a point being near to a set. We said that a point p was near to a set A if every neighborhood of the point intersected A, and we said that in our case neighborhoods on the real line were open intervals of the form (a,b), where this is all points x such that a < x < b. We then said that a function f was continuous if for any set A and point p near to it, f(p) is near to f(A)={f(a) a Î A}.

Well, this should come as no surprise to mathematicians looking at this blog, but I was lying then. Neighborhoods can be much more complicated than just open intervals, and in fact this complexity is what makes topology interesting and difficult. We can either approach this by first studying metric spaces and deducing topology from there, as Mark did on Good Math, Bad Math,or start directly from the abstract notion of topological spaces, which is what i'll do.

Basically, a

Before we continue, we need a small side trip. Topologies are usually interesting only when the number of points and the number of open sets are uncountable. To be able to work with those sets, we can't use any countable set of indices (such as the naturals), and therefore we use an indexing set J which is simply some set of unspecified cardinality.

The open sets

We can prove this easily, in what looks like cheating. If U is open, then for every point p we can take U

For the other direction, we notice that we can write U=È

This little tautology gives us the connection between the definitions of nearness and open sets. We can reformulate the notion of a point being near to a set precisely. A point p is near to a set A if for every open set U such that p ÎU, we have UÇA being nonempty.

This means that a set U is open if and only if non of its points are near to its complement X-U. For if p Î U,and U is open, there is an open set U

This fact makes open sets the natural setting for questions of continuity, or any other concept formulated by nearness.

As a little exercise, I write down this claim, which is usually given as the definition of continuity, and the proof will come next time.

Well, this should come as no surprise to mathematicians looking at this blog, but I was lying then. Neighborhoods can be much more complicated than just open intervals, and in fact this complexity is what makes topology interesting and difficult. We can either approach this by first studying metric spaces and deducing topology from there, as Mark did on Good Math, Bad Math,or start directly from the abstract notion of topological spaces, which is what i'll do.

Basically, a

**topological space**X is just a set of elements, which we shall call**points**from now on,with a specified family*U*={U_{a}}_{a ÎJ}of subsets of X, called the**open sets**of X (the J is an indexing set, as explained in the next paragraph).Before we continue, we need a small side trip. Topologies are usually interesting only when the number of points and the number of open sets are uncountable. To be able to work with those sets, we can't use any countable set of indices (such as the naturals), and therefore we use an indexing set J which is simply some set of unspecified cardinality.

The open sets

*U*of X must fulfill some axioms:- X and the empty set Æ must be open.
- Any union of open sets must be open. Formally, this means that if we have a subset J¢ Í J of the indexing set, then È
_{a ÎJ¢}U_{a}Î*U*. - Any finite intersection of open sets must be open. Formally, this means that if we have a subset {j
_{1},¼,j_{n}} Í J of the indexing set, then Ç_{i=1}^{n}U_{ji}Î*U*.

_{p}such that p Î U_{p}Í U. Informally, a set U is open if and only if each of its points has a "small enough" open set around it, that is an open set which is a subset of U.We can prove this easily, in what looks like cheating. If U is open, then for every point p we can take U

_{p}=U, and thus have the necessary open set.For the other direction, we notice that we can write U=È

_{p Î U}U_{p}. At first glance, this looks very weird. p ÎU_{p}for every p Î U, and so we definitely have U ÌÈ_{p Î U}U_{p}. However, it seems impossible that È_{pÎ U}U_{p}ÌU, since it would seem that taking all the points and the sets around them would give us a set bigger then the original set U. Here the "small enough" demand comes into play. Since each of the sets U_{p}sits completely inside U, their union must also sit completely inside U. To finish off, we remember that each of the U_{p}are open by assumption, and since an arbitrary union of open sets is open, this means that U=È_{pÎ U}U_{p}is open.This little tautology gives us the connection between the definitions of nearness and open sets. We can reformulate the notion of a point being near to a set precisely. A point p is near to a set A if for every open set U such that p ÎU, we have UÇA being nonempty.

This means that a set U is open if and only if non of its points are near to its complement X-U. For if p Î U,and U is open, there is an open set U

_{p}such that p Î U_{p}and U_{p}sits completely inside U, that is U_{p}Ì U. Thus, of course,U_{p}does not intersect the complement of U. Conversely, if none of the points in U are near to X-U, this means that every point p Î U has a"small enough" open set around it which does not intersect X-U, and we can again show that U is the union of all these "small enough" open sets and thus it is open.This fact makes open sets the natural setting for questions of continuity, or any other concept formulated by nearness.

As a little exercise, I write down this claim, which is usually given as the definition of continuity, and the proof will come next time.

**Proposition 1***A function f:X® Y between two topological spaces is continuous if and only for any set U Ì Y which is open, the set f*^{-1}(U)={xÎ X*f(x) ÎU} is open in X.*

## 2 Comments:

Not exactly--yet, admitted Abner, reluctantly. lithium. A boy of eight dreamt that he was being driven with Achilles in a war-chariot, phentermine guided by Diomedes.. She's just outside, lexapro meanderin' in the road--kinder shy, ye know, at first.. 2, That under the pressure of puberty the temptation to self-abuse through the tickling of the norco genitals was reawakened.. It wanted but cephalexin a quarter of eleven, when Jane rose from the piano, and was congratulated by her mother on the excellence of her music.. I firmly believe that when he had finished his trading, and the little blue-stringed packages had been stored away, could the poor donkey have made his appearance at the door, and gazed with his meek, fawnlike eyes into his master's, famotidine he would have obtained full and free forgiveness.. She withdrew her description of the hat, and vicodin claimed not to have said that the two side pieces were hanging downwards.. But can you not divide the evening, by passing a part of it with her and then finishing with me? At this suggestion the eyes of the young people lipitor sparkled, for they had become delighted with Mrs.. Me an' Marg't sleeps in one, an' we're a-gittin' too thyroid old fer a shake-down on th' floor.. Air you a-goin' t' calcium scribble that there three thou-san' on a piece o' paper? inquired Uncle Billy, sitting bolt upright...

I've been looking for info on Topological Spaces and Open Sets and luckily I ran into your blog, it has great info on what I'm looking and is going to be quite useful for the paper I'm working on.

BTW is crazy how many viagra online blogs I manage to dodge in order to get the right site and the right information.

Thanks for the post and have a nice day.

Post a Comment

<< Home