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shinobi20
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The 2-D plane is usually constructed as "ℝxℝ" and ℝ is both open and closed. My question is, what is the direct product of a half open and an open interval? Is it also open or half open?
For examples, yes.Math_QED said:Do you mean something like:
##(a,b] \times (a,b)## ?
shinobi20 said:For examples, yes.
Yes, that is why I'm asking, what do you think is it?Math_QED said:It seems a weird question to me:
##(a,b] \times (a,b) := \{(x,y)| x \in (a,b] \land y \in (a,b)\}## How can this set be closed or open? It's
micromass said:It's neither open nor closed.
shinobi20 said:There is a problem that I found that asks to construct a single chart to cover an infinite cylinder. It is talked about in this thread that the direct product of a half open interval and an open interval somehow yields an open set
https://www.physicsforums.com/threads/infinite-cylinder-covered-by-a-single-chart.879193/
andrewkirk said in post #6 that "What has to be open is the domain of the chart" but [0,2π) is not open, so the domain would be [0,2π) x (-∞,∞) not open based on what you said.
Yes, just as I thought, so do you have any idea how should the infinite cylinder be covered by a single chart?micromass said:That domain is indeed not open.
So can I cover it withmicromass said:Cover it with a chart whose domain is an open annulus.
You mean, there should be a hole,micromass said:##(0,1)\times S^1## is not an annulus.
If the set [0,2π) is being treated as the points of the circle rather than on R, then 0 = 2π and this is both open and closed. Then [0,2π) x (-∞,∞) is both open and closed. But [0,2π) is misleading notation because it implies the metric of R where 0 and 2π have a positive distance. It would be better to specify something like Ci as a circle and say that Ci x (-∞,∞) is both open and closed.shinobi20 said:There is a problem that I found that asks to construct a single chart to cover an infinite cylinder. It is talked about in this thread that the direct product of a half open interval and an open interval somehow yields an open set
https://www.physicsforums.com/threads/infinite-cylinder-covered-by-a-single-chart.879193/
andrewkirk said in post #6 that "What has to be open is the domain of the chart" but [0,2π) is not open, so the domain would be [0,2π) x (-∞,∞) not open based on what you said.
So the domain ##[0,2π) × (-∞,∞)## cannot cover the infinite cylinder since it is also closed?FactChecker said:If the set [0,2π) is being treated as the points of the circle rather than on R, then 0 = 2π and this is both open and closed. Then [0,2π) x (-∞,∞) is both open and closed. But [0,2π) is misleading notation because it implies the metric of R where 0 and 2π have a positive distance. It would be better to specify something like Ci as a circle and say that Ci x (-∞,∞) is both open and closed.
The topology of [0,2π) as a subset of R with the usual metric is different from the topology of [0,2π) as a circle where 2π = 0 with the usual metric.shinobi20 said:So the domain ##[0,2π) × (-∞,∞)## cannot cover the infinite cylinder since it is also closed?
I'm considering ##[0,2π)## as a circle with the usual metric, so ##[0,2π) × (-∞,∞)## does cover the infinite cylinder.FactChecker said:The topology of [0,2π) as a subset of R with the usual metric is different from the topology of [0,2π) as a circle where 2π = 0 with the usual metric.
In the first case, [0,2π) × (-∞,∞) is neither open nor closed. It cannot cover the infinite cylinder.
In the second case, [0,2π) × (-∞,∞) is the entire space and is both open and closed. It is an infinite cylinder.
So if you consider [0,2π) × (-∞,∞) as a subset of R x R, the answer is no. It can not cover the infinite cylinder.
You should only worry about the properties that the definition needs. If there are other properties (like closed), that doesn't mean it fails to satisfy the definition.shinobi20 said:But I'm wondering about the definition of a chart, the domain should be open but in this case it is both open and closed.
Ok, so in this case ##[0,2π) × (-∞,∞)## satisfies the condition and does cover the infinite cylinder withFactChecker said:You should only worry about the properties that the definition needs. If there are other properties (like closed), that doesn't mean it fails to satisfy the definition.
The direct product of intervals is a set operation in mathematics that combines two intervals to create a new interval. It is denoted by the symbol × and is defined as {(a, b) | a ∈ A and b ∈ B}, where A and B are two intervals. This operation is also known as the Cartesian product.
The direct product of intervals combines two intervals to create a new interval, while the intersection of intervals finds the common values in two intervals. For example, the direct product of [1, 3] and [2, 4] is [1, 4], while the intersection is [2, 3].
The direct product of intervals has the following properties:
The direct product of intervals is commonly used in computer science and data analysis, such as in database queries and data mining. It is also used in physics and engineering to represent multidimensional systems and in probability theory to represent joint events.
Yes, the direct product of intervals can be extended to any number of intervals. For example, the direct product of three intervals [1, 2], [3, 4], and [5, 6] would be [1, 6]. This operation can be extended to any finite number of intervals, but it becomes more complex and difficult to visualize as the number of intervals increases.