General topology - 2016-12-04

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4:26 AM

in Mathematics, 12 mins ago, by Secret

Hey guys have a topology conceptual question here:
Suppose I have a 3 element set $\{a,b,c\}$ wit the following topologies:
$$\tau_1=\{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$$
$$\tau_2=\{\emptyset,\{b\},\{a,b\},\{b,c\},\{a,b,c\}\}$$
$$\tau_3=\{\emptyset,\{a,b,c\}\}$$

Is the following interpretation correct (omitting the description about the emptyset):
For $\tau_1$ all elements form an open set, and no two elements are considered near to each other in this topology
For $\tau_2$, b form an open set, while a,c and a,b form two other open sets. Therefore a and c are c…

4:40 AM

in Mathematics, 1 min ago, by Secret

@MartinSleziak O sorry I made a typo, I mean a,b are close together and b, c are close together but not a, c

5:27 AM

in Mathematics, 6 mins ago, by Martin Sleziak

Anyway, since you say that you want to somehow define when two points are in some sense close to each other, wouldn't defining points to be closed if they are not separated be something similar to what you have in mind.

in Mathematics, 3 mins ago, by Secret

@MartinSleziak I am suspecting that, I kinda have this thought yesterday when maths chat talked about the padic numbers. (please wait when I upload the pics to see if I understood correcrly, I am too slow in typing)

in Mathematics, 3 mins ago, by Secret

user image

I have copied a few messages from the other chat room - if somebody peeks in this one, they will see what this conversation is about.

I hope I posted relevant stuff.

I will copy also definition: "Two points x and y are separated if each of them has a neighbourhood that is not a neighbourhood of the other; that is, neither belongs to the other's closure." en.wikipedia.org/wiki/Separation_axiom

Both $\tau_1$ and $\tau_3$ are easy.

In discrete topology any two points are separated.

In indiscrete topology no points are separated.

@Secret In the main chatroom you said $$\tau_2=\{\emptyset,\{b\},\{a,b\},\{b,c\},\{a,b,c\}\}.$$ Is this what you mean by the second picture?

I am asking mainly because you did not plot the set $\{a,b\}$ and $\{c,b\}$ in the picture - both of them are open.

If this is definition of $\tau_2$ then a and c are separated. The points a and b are not separated, neither are c and b.

Currently, inspired from the discussion of the tree like topology of p-adic numbers yesterday, my current conception of topology is like what is shown on the diagram, thinking that any points that are not separated are circled, as if they are located within sets that are not disjointed:

I think fargle got it right that I rely on the notion of separable points (both in the formal sense in terms of neighbourhoods, which also happens to somewhat coincide with the daily life notion of "two things close to each other") to make sense of general topology.

$\tau_1=\{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$

So $\tau_1$ is discrete?

yes

In discrete space, closure of a point is singleton: $\overline{\{x\}}=\{x\}$.

In your case closer of $\{a\}$ is $\{a\}$.

More clarification of the diagram: The circled stuff represent a set that contains elements that are not separated, rather than open sets given by the topology

Ah I see

5:38 AM

Notice that $b\notin\overline{\{a\}}$. Since since $b$ has a neighborhood $\{b\}$ which misses $a$.

is the closure of the whole set the whole set, which means the whole set is always clopen?

@Secret I am used to pictures where open sets are drawn - like most of the pictures here. That's why I did not understand the diagram.

@Secret Yes. Both $\emptyset$ and X are clopen in X. For every topological space.

If there is a clopen set different from $\emptyset$ and $X$, then means $X$ is not connected.

I see

Manwhile bolbeteppa also gave some useful insights. Still digesting though

in Mathematics, 5 mins ago, by bolbteppa

@Secret In $\tau_1$, $\{ a,b,c \}$ are close to each other, up to some given 'error', while $\{ a,b \}$ are also close to each other in a sense different to how they are close to $c$, i.e. $a$, $b$ and $c$ can approximate one another up to some given error, which is how you interpret $\{ a,b,c \}$, but $a$ can also approximate $b$ up to some other given error, which is how you interpret $\{ a,b \}$, where the notion of error is unspecified, e.g. in a metric space you can use a distance function

in Mathematics, 2 mins ago, by bolbteppa

@Secret in $\tau_2$, $b$ is a number you know exactly, like, say, a rational number, that can be written down, while $a, c$ might be irrational numbers, you can't write down and don't know the exact value of, e.g. $\sqrt{2}$, so, $\{a,b,c\}$ means we interpret $b$ as close to $a$ and $c$ up to some given level of error, while $\{b,c\}$ means that $b$ and $c$ are close to each other up to some other given level of error and hence approximate one another, same with $\{a,b\}$,

in Mathematics, 2 mins ago, by bolbteppa

while $\{b\}$ means we can approximate $b$ with all elements in that set, namely $b$, it is close to itself up to some given level of accuracy.

in Mathematics, 1 min ago, by bolbteppa

Said another way, topology is the language of approximation

but I think combined with both of your responses, I know what the empty set and whole set are doing in the topology

now I just need to digest the tau-2 case... this might take some time before it all clicks

BTW if one of your interest is to gain some intuition, you might browse a bit through questions tagged intuition+general-topology to see whether some answer there will be helpful.

And you might find something useful also on MathOverflow, Quora, Reddit and other sites. I guess that if you try a few reasonable searches in google, you might find a few such posts. For example, I tried to search for topology open sets intuition and at least some reasonably looking links are among the results.

Searching for topological space intuition returns several posts which - judging by the tiles - look reasonable.

Of course, it will take some time to choose which ones of them are worth reading. And then actually read the answers. (Here on SE, votes might help with choosing the ones which are likely among the best.)

5:55 AM

What is a topology intuitively? Formally it is a set of subsets of another set that satisfies some properties, but intuitively?

@DHMO I do not have better answer that: a) You can look at posts from other links. b) It depends on context you are working with.

@MartinSleziak For example, $\tau_2=\{\varnothing,\{b\},\{a,b\},\{b,c\},\{a,b,c\}\}.$

user image

I fail to see how the middle diagram is an intuition for $\tau_2$.

that diagram is inaccurate, don't use it

it failed to acocunt for the two eleemnt open sets

For example, in some situations, instead of open sets it is much easier to work with convergence of nets. So in this case the intuition would be: "This is simply something like the convergence we are used to, but more general."

as bolbteppa implicitly point out

5:57 AM

Then how can one build an intuition for $\tau_2$?

@DHMO When you ask for "intuition", did you meant that questions as a questions about topological spaces in general or something along these lines: "I have this particular space. How can I understand it better."

@MartinSleziak the latter

And speaking of diagrams, it is more about visualization/drawing topological spaces.

For me, to build an intuition on topology is to understand why a topology is defined to be closed under union and intersections, and what is the "missing bit" for a collection that is not a topology (e.g. throwing out {b} in $\tau_2$. I think Martin, Fargle and Bolteppa have gave the answers I need (plus my own background reading in MSE Munkres etc. I just need to synthesise the info together and that takes time which is why I have nto chat for a while

@Secret why must they be closed under union and intersections?

6:01 AM

I am still processing...

Basically $\{\{a,b\},\{b,c\}\}$ can generate $\tau_2$, which I think is a good way to understand it.

expect no answer for me from at least 10 mins

Again, this depends a lot on the situation and for some topological spaces it would be very difficult to draw a good picture. But for finite spaces I am used to pictures where either all open sets are pictured, or some sets which generate the topology. (The latter is usually better.) Like the pictures of topologies with three points here: google.com/search?q=topologies+three+elements&tbm=isch

user image

$\tau_1$ is the bottom right diagram
$\tau_2$ is the top right diagram
$\tau_3$ is the top left diagram

Similar pictures might be helpful in some cases, like order topology. But in other cases drawing a picture like that would be impossible.

Although for many topologies which are modification of the usual topology on $\mathbb R$ or $\mathbb R^2$, pictures showing basic open sets might be useful. Like Moore plane or Sorgenfrey line.

6:53 AM

Synthesis results:
(For convenience neari for = integers denote the different abstract/unspecified notions of nearness/next to each other)

$$\tau_1=\{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$$
Discrete topology, all a,b,c are separated from each other, but near1, near2, near3 to itself. every two pair of points are near4, near5, near6 to one another, and all point are near7 to all others. There is also no point are near8 to each other (emptyset case)

$$\tau_2=\{\emptyset,\{b\},\{a,b\},\{b,c\},\{a,b,c\}\}$$

typo: neari for i = integers

@Secret What on earth is near1 near2 etc?

different notions of nearness, basically adapting from bolteppa notion of errors

what does near1 mean?

given near1 and near2. If we happened to work in a metric space, then near1 corresponds to some distance e.g. 1, in order for two points to be considered next to each other, and near2 corresponds to some distance e.g. 45, in order for two points to be considered next to each other

so the various labelled near$_i$ denotes something simialr except more abstract, e.g. it might be satisfying some properties before two points are considered next to each other

Or more clearly, near$_i$ means the points are not separated under some property i

So they can be separated under property 1 but not under property 2

7:15 AM

As I mentioned, I'll have to do other stuff, so I am going to leave the chat room now. Before leaving I will mention one more thing which perhaps might be interesting for you (or maybe not).

If you would think about when two sets are near each other (rather two points), there exist structures called nearness spaces and proximity spaces. See also this question: Whatever Happened to Nearness Spaces?

However, I am not familiar with either of these two concepts. So I cannot tell you about them much more than I have heard the buzzwords and you can probably find some literature about them.

ok thanks, I'll have a check

7:37 AM

0

Q: Normal covering space and resulting isomorphisms

When studying for an exam, I came across this question in Hatcher, and it has stumped me. Let $G$ be a group with normal subgroup $N$. Show that there exists a normal covering space $\tilde{X} \rightarrow X$ with $\pi_1(X) \approx G, \pi_1(\tilde{X}) \approx N$, and deck transformation group $G...

general-topology algebraic-topology

1 hour later…

8:39 AM

For any guys who may be reading: I think I have found something that can express precisely the above two questions:

Closeness (mathematics)

Closeness is a basic concept in topology and related areas in mathematics. Intuitively we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a metric space where a notion of distance between elements of the space is defined, but it can be generalized to topological spaces where we have no concrete way to measure distances. Note the difference between closeness, which describes the relation between two sets, and closedness, which describes a single set. The closure operator closes a given set by mapping it to a closed set which contains the...

I also read about the proximity space (which basically behave like the closure operator definition generalised to between sets), and the closure operator definition of topology. I understood how they are equivalent to the open set definition.

Now I just need to figure out how to recover the notion of closeness from the open set definition in order to work out why the "not a topology" case is not useful

user image

I think using this I have figured it out. My analysis is as follows:

9:08 AM

Suppose there is a set $X=\{a,b,c\}$ and a topology $\tau_x=\{\emptyset,\{a\},\{b\},\{a,b\},\{a,b,c\}\}=\{\emptyset,A,B,C,D\}$. From this topology and the notion of closeness given in the wikipedia link we can deduce the following:
1. no points are close to the empty set
2. $a\in A$ thus only a is close to A
3. $b \in B$ thus only b is close to B
4. $a,b, \in C$ thus both a and b are close to C
5. $a,b,c \in D$ thus all a,b,c are close to D.

Why the closure under intersections is necessary in the axiom:

@Secret What is the difference between "p is close to A" and "p in A"?

p can be not in the open set A, but inside the closed set that contains A

(spelt out in the "generalised definition" in the wikipedia link)

@Secret Can you demonstrate closure with finite elements?

10:06 AM

A set in the topology is closed if its complement is in the topology. In particular the whole set and empty set are always clopen
$\tau_1=\{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$
$cl(\{a\})=\{a\}$
$cl(\{b\})=\{b\}$
$cl(\{c\})=\{c\}$
$cl(\{a,b\})=\{a,b\}$
$cl(\{b,c\})=\{b,c\}$
$cl(\{a,c\})=\{a,c\}$

$\tau_2=\{\emptyset,\{b\},\{a,b\},\{b,c\},\{a,b,c\}\}$
$cl(\{b\})=\{a,b,c\}$
$cl(\{a,b\})=\{a,b,c\}$
$cl(\{b,c\})=\{a,b,c\}$
NB $\{b\}^C=\{a,c\}\not\in \tau_2$, $\{a,b\}^C=\{c\}\not\in \tau_2$, $\{b,c\}^C=\{a\}\not\in \tau_2$. Therefore the only closed sets are the whole…

10:30 AM

Attempt at intuition: Under the open set definition, the open sets of a topology determines what collection of sets of points (e.g. $\emptyset$,{a},{a,b},{b,c},{a,b,c} etc.) these sets are close to. If a set is closed in the topology, it means given a collection that are close/near to this set, it is the only set this particular collection is close/near to
(and once step outside of this set into the complement, only points from a different collection are close/near to this complement). If the complement of an open set is not in the topology, it means we cannot say whether this particular co…

However suppose I ask what is the complement of {a} in $\tau_2$, then I get {b,c} which is in $\tau_2$, thus {a} is closed. But in $\tau_2$ there is no notion of "only a is close to the set {a}" because {a} is not in the topology. Therefore, what does it mean when it is found that {a} is closed in $\tau_2$?

3 hours later…

user131753

1:47 PM

I have been thinking sometime regarding an analogue of the concept of homomorphism in topology.

user131753

Although Brain M. Scott in this question of mine answered it, I was wondering what exactly is wrong with my initial idea of an open injective map being a topological analogue to homorphism.

user131753

As you can see in the chain of comments, I think that it must be a very trivial thing that I am missing because as he said "[o]pen injective maps simply don’t behave at all like algebraic homomorphisms". Can you give me some idea as to the reason(s) behind it @MartinSleziak?

@user170039 Perhaps I'll have a look at it sometime in the future when I have more time. But in general I am not too good in metamathematical and big picture questions.

But let's hope that somebody else who peeks into this room will be able to tell you something about this.

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