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I can't understand the differences between a non-trivial topology and a trivial one. Whuat's the meaning of "non-trivial" topology? Is there a link with connection's properties?

For example, could we say that a moebius strip has a "non-trivial" topology while an ordinary strip has a trivial one?

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The trivial topology on a set $X$ is usually defined to be the collection $\{ \emptyset, X \}$. Any other topology on $X$ is nontrivial. Remember a given space can have several or infinitely many different topologies put on it. –  grantfgates Jul 7 '13 at 11:13
    
In particular, the only type of continuous function from a trivial topology are the constant maps. –  Thomas Andrews Jul 7 '13 at 11:20
    
The ordinary strip and the Möbius strip both have non-trivial topologies. It really has nothing to do with visual intuition. –  Thomas Andrews Jul 7 '13 at 11:23
    
Is there a link between a non-trivial topology and the genus? –  xino Jul 7 '13 at 20:45

2 Answers 2

up vote 3 down vote accepted

A topology $T$ on a set $X$ is the set of open sets (more precisely: the sets we will afterwards call open sets when working with this topology $T$; openness is no absolute and inherent property of a subset of $X$), that is a set of subsets of $X$ such that

  1. $X\in T$
  2. If $I$ is an index set and $U_i\in T$ for each $i\in I$, then $\bigcup_{i\in I}U_i \in T$
  3. If $U,V\in T$ then $U\cap V\in T$.

Hence for any nonempty set $X$ there is one topology, where checking conditions 1., 2., 3. is trivial, i.e. does not require "any" computational effort: $T=\{\emptyset,X\}$. That's why this topology is called the trivial topology on $X$ (also: the indiscrete topology). As a matter of fact, verifying conditions 1., 2., 3. is also trivial if one choses $T$ to be the powerset of $X$, however that topology runs by the name of discrete topology. This is so even hough the verification of the above conditions is "even more trivial" in this case: If all sets are in $T$ anyway, there is nothing to check. Rule of thumb: If ever you have two choices for naming trivial objects, the "smaller" one wins.

Compare this to other cases where objects are called trivial, such as the trivial group having only a single element $e$ with composition $e\cdot e=e$ (often occurring in phrases such as "the kernel of ... is trivial")

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"the set of open sets" is a bit vague - it implies that there might be only one such set. It's a bit backwards to use the word "open" before you've defined topology. –  Thomas Andrews Jul 7 '13 at 11:25
    
@ThomasAndrews Well, it was intended as motivation only ... –  Hagen von Eitzen Jul 7 '13 at 11:28
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Yeah, my point is that motivationally, I've found people confused by this definition, because they already "know" open sets from analysis, and think they are fixed externally, rather than defined by $T$ (so that they later don't understand two different topologies on the same set, for example.) –  Thomas Andrews Jul 7 '13 at 11:31

A trivial topology on a set is a topology that, in some sense, is not interesting from the point of view of topology. There are two candidates for topologies on a set $X$ that are trivial. One is $\{X,\emptyset\}$ and the other is $\mathcal P(X)$. It is the former that is called 'the trivial topology'. The latter is also known as the discrete topology, and then the former is called the indiscrete topology. A set $X$ with one of those topologies is essentially just the set $X$. Since 'trivial' is in the eye of the beholder, it may be better to stick to discrete and indiscrete (though in topology 'trivial topology' is synonymous with 'indiscrete topology').

Discrete and indiscrete topologies (and by the way both the Moebius strip and an ordinary strip, as sets, support both the discrete and indiscrete topologies, but when one refers to them as topological spaces the topology assumed is neither) can be characterized in various ways and have various properties. A space $X$ is discrete iff every function $f:X\to Y$ to any other space $Y$ is continuous. Dually, $X$ is indiscret iff every function $f:Y\to X$ from any other space $Y$ is continuous. An indiscrete space is always connected, while a discrete space is connected iff it is a singelton. This might give you some intuition as to what they 'look like'.

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I have never heard the discrete topology called "trivial," and I think this answer would lead one to believe that some people do. The discrete topology might be the other "obvious" topology, but "trivial" doesn't always mean "obvious." –  Thomas Andrews Jul 7 '13 at 11:35
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Also, I think the "Möbius strip" as a set doesn't exist, it is a set with a topology already, not a set without topology that you can slap any old topology onto - if you had a different topology, it wouldn't be a Möbius strip. –  Thomas Andrews Jul 7 '13 at 11:38
    
@ThomasAndrews I removed some words to emphasize that it is the indiscrete topology that is referred to as trivial in topology. And thanks for the second comment. –  Ittay Weiss Jul 7 '13 at 11:43

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