Trivial topology on an arbitrary set?

[muppet]

I've just started looking at Rudin (Real and complex analysis, the big one) in which he offers the following definition (1.2):

a)A collection $$\tau$$ of subsets of a set X is said to be a topology in X if $$\tau$$ has the following three properties:
i) $$\emptyset \in \tau$$ and $$X \in \tau$$
ii) If $$V_{i}\in\tau$$ for i=1,$$\cdots$$,n, then $$V_{1} \cap V_{2} \cap \cdots \cap V_{n} \in \tau$$
iii) If {$$V_{a}$$} is an arbitrary collection of members of $$\tau$$ (finite, countable or uncountable) then $$\bigcup_{\alpha}V_{\alpha}\in \tau$$.

b)if $$\tau$$ is a topology in X, then X is called a topological space, and the members of X are called the open sets in X.

(Apologies for the random superscripts, I'm not sure why LaTeX felt they were necessary).

By considering the set {X, $$\emptyset$$} doesn't this make every set a topological space? The first condition is obviously satisfied, the union of the empty set and any other set X is X, and the intersection of X with the empty set is the empty set... right?

 

[quasar987]

Right. That is called the indiscrete topology. Another obvious topology possible is the discrete topology, in which $\tau=2^X$ (the set consisting of all subsets of X).

 

[muppet]

So I'm not going mad then... thanks!

 

[HallsofIvy]

Well, now, we can't speak to that!

 

[jostpuur]

Hello muppet, I'm curious about one thing. Did you immediately understand, that when we speak about a set being open, it is always meant, that it is open with respect to some topology?

It's just that I didn't understand this, when I read the Rudin for the first time. I thought that a set $$A\subset X$$ is called open, if there exists a topology $$\tau\subset\mathcal{P}(X)$$ so that $$A\in\tau$$. By choosing $$\tau=\mathcal{P}(X)$$ it then followed that every set is always open. It took me more than a week to find out that openess is always defined with respect to some topology, and during this time, I was informed that I was stupid because I hadn't understood topology (traumatic memories about mathematics...)

 

[HallsofIvy]

Oh, please tell me that it was not a teacher who said you were stupid!

 

[jostpuur]

No, he was one physicist, whom with I communicated through the IRC. That's also why this wasn't strictly negative experience with mathematics. And one reason why I started getting interested in finding other places to talk about mathematics and physics. This happened before I had found PF

 

[muppet]

I think you've saved me some head-scratching there- thanks! I'd picked up that an open set belonged to some particular topology but hadn't fully followed that to its conclusion yet; I'll admit to a couple of raised eyebrows when he "defined" a set as being open or closed (which I now see as really defining a particular topoology on some space).
I'm viewing Rudin as a sort of project- 400 pages, about 4 pages a day, and as I won't do some every day then I'll try and be finished by Christmas. Of course, my resolution will be tested once term starts
By the way, I should have specified a very particular definition of mad above- one whose trains of thought pertaining to the platonic world in which mathematical objects reside are derailed by some delusion foreign to that world. For the record, I'm entering my 3rd year at a UK university, and Rudin has no relevance to a course I'll be taking this or probably next year. Does that qualify me as being suitably barking for you to consider speaking to me again?

 

[jostpuur]

The topology book which I have introduces the concept of open sets like this:


The members of $\mathcal{T}$ are called open (more precisely $\mathcal{T}$-open) sets.


That is a good way to say it. Even if you are not going to call open sets usually $\mathcal{T}$-open sets, it is good to know that they could be called that way. This way the student immediately understands what is meant by the concept.

 

[math108]

I am at the end of three weeks in this semester, and I had the same problem. I will have my first exam the week after next week and you both help me tremendously.

 

[Hurkyl]

(Apologies for the random superscripts, I'm not sure why LaTeX felt they were necessary).

Use [itex] instead of [tex]. (And similarly for the closing tag) The former is better formated for inline LaTeX. In fact, the two types of tags directly correspond to $...$ and \[...\].

 

[shoehorn]

If you're confused about the concept of an open set, all you need to remember is that the original motivation for calling these sets "open" is due to the canonical example of an open set being provided by the usual topology on $\mathbb{R}$. More specifically, the usual topology on $\mathbb{R}$ consists of all open intervals $(a,b)$ and their unions; the fact that these open intervals form a topology then gave rise to the practice of referring to any of the sets in a topology as open sets.

(Of course, another reason why confusion exists regarding open sets is that a topology $\mathcal{T}$ on a set $X$ necessarily includes both $X$ and $\emptyset$. Unfortunately, the way in which we define closed sets implies that $X$ and $\emptyset$ are both open and closed. Go figure.)