An open set can be a closed interval?

  • #1
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I am confused with some basic definitions in general topology.

Topological space is defined as a set ##X## and a collection ##T## of its subsets that satisfies some general properties (which I will not list here, but I will assume that the reader knows them). But such a very general definition of a topological space does not seem to capture the intuitive idea that topology is about distinguishing objects that cannot continuously be transformed into each other (e.g. coffee cup and torus are topologically the same). What's the motivation for such a general definition of a topological space?

Even more confusing is that the sets in ##T## are called open sets. Why are they called so, given that some of them can be closed intervals?
 

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  • #2
Ssnow
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Even more confusing is that the sets in ##T## are called open sets. Why are they called so, given that some of them can be closed intervals?

In order to have a topology you must specificate what are the "open sets" well, this is a convention because you can specify a topology also declaring what are the "closed sets", in this case axioms changes a bit ...

Ssnow
 
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In order to have a topology you must specificate what are the "open sets" well, this is a convention because you can specify a topology also declaring what are the "closed sets", in this case axioms changes a bit ...
Fine, but I would like to know why those sets are called "open". Perhaps for some historical reasons? For instance, maybe mathematicians in the past used a less general definition of topological space, in which sets in ##T## were all like open intervals/disks/balls?
 
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Open sets in a topology have the same properties that open balls in the standard metric in Euclidean space do. They do not have the same properties that closed sets in Euclidean space have.

I'm general open sets in a topological space as generalizations of open balls, and I think a lot of problems become a bit more intuitive if you think about it that way.
 
  • #5
Ssnow
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I think open sets are "conceptually minimal" respect closed sets in the sense that there is common procedure to pass from an open set to a closed set, it is called the "clousure operation", it is like an assertion that can be free or closed:

## x+y=1 ##

is open

##\forall x,y(x+y)=1##

is closed beacuse there are no variables free ... I think in topology is less or more the same ... "Philosophically you close an object (add a boundary), it is no interesting to open it ... " I don't know if I have clarify, my it is only a personal opinion ...

Ssnow
 
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Open sets in a topology have the same properties that open balls in the standard metric in Euclidean space do.
In what sense do they have the same properties? What these properties are?
 
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In what sense do they have the same properties? What these properties are?

I mean, all the axioms of open sets are satisfied by open balls. They're not satisfied by closed sets. The one about infinite unions returning another open set is the key one I think, this is very vaguely describing some sort of limiting process (think about what an infinite union of intervals center at 0 looks like)
 
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It is not a good idea in my opinion to think of open balls (intervals) as a basis of open sets. This will leave you almost certainly with (Euclidean) metric spaces. That's fine as long as you deal exclusively with those spaces, but a mean pit if you deal with topological spaces in general. Intuition is a bad advisor when it comes to topology.

Dieudonné said that Cantor incidentally brought topology to life when he was investigating the set of points whose values can be changed without changing the Fourier expansions of functions in 1870.

His statement is of course far more detailed, but I'm too lazy to type it from the book. In any case, we should distinguish between general topology and algebraic topology. The example with the cup is algebraic topology, which investigates algebraic invariants of topological objects. General topology is in the end advanced set theory (duck and cover). However, metric spaces are nice, but far from typical for topologies.
 
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  • #9
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I am confused with some basic definitions in general topology.

Topological space is defined as a set ##X## and a collection ##T## of its subsets that satisfies some general properties (which I will not list here, but I will assume that the reader knows them). But such a very general definition of a topological space does not seem to capture the intuitive idea that topology is about distinguishing objects that cannot continuously be transformed into each other (e.g. coffee cup and torus are topologically the same). What's the motivation for such a general definition of a topological space?

Even more confusing is that the sets in ##T## are called open sets. Why are they called so, given that some of them can be closed intervals?
Functional analysis is based on analytic topology, which is the idea of defining the minimal structure on a set in order to have the concept of a continuous function. The starting point is the alternative definition of a continuous function: that the pre-image of every open set is an open set (*). This frees continuity from any concept of distance (as you have with a metric space). To generalise the concept of continuity, therefore, you don't need a metric or norm or inner product or anything like that. But, you do need to establish the requisite properties of open sets that support the generalisation of continuity, using the above definition.

The properties defined in a topology turn out to be sufficient (although they themselves are perhaps suprisingly minimal).

Algebraic topology branches off from analytic topology and is a different beast altogether!

(*) Exercise: prove this is equivalent to the epsilon-delta definition for real-valued functions.
 
  • #10
wrobel
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A closed interval is closed as a subset of ##\mathbb{R}## and open as a topological space with inducted from ##\mathbb{R}## topology. By definition any topological space is open and closed simultaneously.
 
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  • #11
pasmith
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Most of the confusion disappears once you accept that "open" is not the logical negation of "closed".
 
  • #12
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Point set topology generalizes topological properties from Euclidean space : Compactness, Completeness, etc. , into more general spaces. It happens to be that open sets serve well to describe these properties and to describe other important ones such as continuity : a function is continuous if the inverse image of an open set is open. Ultimately, maybe somewhat circular, Topology seeks to classify all spaces up to homeomorphism. Algebraic Topology is an aid for this goal : it attaches an algebraic object to each topological space accirding to " functorial" properties. But it's not precise-enough to distinguish beyond Homotopic spaces. Open sets S are those for which each point has a (basis) neighborhood completely contained within S . Please ask if I was not clear.
 
  • #13
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A closed interval is closed as a subset of ##\mathbb{R}## and open as a topological space with inducted from ##\mathbb{R}## topology. By definition any topological space is open and closed simultaneously.
And there is a result that topological spaces in which there are sets that are both open and closed other than the full space itself, are not connected. General Topology seeks to apply somewhat understandable , " intuitive" concepts from Euclidean space into other spaces. Open sets have been found to be useful in describing and generalizing these properties.
 

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