Why do we specify chart image is open?

  • Thread starter LAHLH
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In summary, when defining a manifold and its charts, one of the conditions specified is that the image \phi(U) is open in R^n. This is important because it ensures that the coordinate chart is defined on an open set, allowing for continuity and the ability to define a metric at a given point. Additionally, this requirement is necessary to correctly define a manifold as something that "looks like" \mathbb{R}^n locally.
  • #1
LAHLH
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Hi,

When we define a manifold, and in particular define what a chart is, one of the conditions we specify is that the image [tex] \phi(U) [/tex] is open in [tex]R^n[/tex]. Why do we specify this?

For example if we didn't specify this and allowed closed balls, then we could cover [tex] S^1 [/tex] by [tex] \theta [/tex] where [tex]\theta \in [0,2\pi) [/tex], and wouldn't need two charts.

I know that open balls and continuity go hand in hand, so I understand if we take [tex] U \subset M [/tex] as an open interval and want to define a continuous map [tex] \phi [/tex] then it must be that [tex] \phi(U) \subset R^n [/tex] is open. But what if our [tex] U \subset M [/tex] is not open, then why do we care if the image is open or not?

Basically why is this part of the definition?
 
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  • #2
I can't think of a good answer right now, so I'll just point out that [itex]f:X\rightarrow Y[/itex] is said to be continuous if [itex]f^{-1}(V)[/itex] is open for each open [itex]V\subset Y[/itex], and said to be open if [itex]f(U)[/itex] is open for each open [itex]U\subset X[/itex].
 
  • #3
Fredrik said:
I can't think of a good answer right now, so I'll just point out that [itex]f:X\rightarrow Y[/itex] is said to be continuous if [itex]f^{-1}(V)[/itex] is open for each open [itex]V\subset Y[/itex], and said to be open if [itex]f(U)[/itex] is open for each open [itex]U\subset X[/itex].


Yep, sorry was being a bit sloppy above.
 
  • #4
LAHLH said:
For example if we didn't specify this and allowed closed balls, then we could cover [itex] S^1 [/itex] by [itex] \theta [/itex] where [itex]\theta \in [0,2\pi) [/itex], and wouldn't need two charts.
But then you wouldn't necessarily have the required continuity at [itex]\theta = 0 [/itex]
 
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  • #5
I think DrGreg's #4 works.

As an alternative way of looking at it, you want to be able to define a metric at a given point P in a chart. Say you want to express the metric as a line element [itex]d\ell^2=g_{ab}dx^a dx^b[/itex]. You need to be able to fit those infinitesimally small dx's inside a neighborhood of P, without leaving the domain of your coordinate chart. If the chart is defined on an open set, then you're guaranteed that they fit. If it's not, then P could be a boundary point, and you would then have at least one dxa that didn't fit inside the chart.

Even if you're not interested in defining a metric -- say you just want to talk about manifolds as abstract topological spaces. The standard definition of a manifold is something that "looks like" (i.e., is homeomorphic to) [itex]\mathbb{R}^n[/itex] locally. If you don't interpret "locally" to mean "on some sufficiently small *open* set," then you don't correctly capture the intended definition of a manifold, as opposed to a manifold with boundary or something else.
 

Related to Why do we specify chart image is open?

1. Why do we need to specify that the chart image is open?

Specifying that the chart image is open allows other researchers to access and reproduce the data and results presented in the chart. It also promotes transparency and accountability in the scientific community.

2. What does "open" mean in the context of chart images?

In this context, "open" refers to making the chart image and its accompanying data and methods freely available to the public. This can include sharing the data and code used to create the chart, as well as any relevant documentation or explanations.

3. How does specifying that the chart image is open benefit the scientific community?

By making the chart image open, other researchers can use the data and methods to verify the results and build upon them for further research. This can lead to new discoveries and advancements in the field.

4. Are there any downsides to specifying that the chart image is open?

Some researchers may be hesitant to share their data and methods due to concerns about being scooped or losing credit for their work. However, the benefits of open science, such as increased collaboration and credibility, outweigh these potential downsides.

5. How can I make sure that my chart image is open?

To ensure that your chart image is open, you can make the data and methods publicly available by publishing them in a reputable journal or on an open access platform. You can also include a statement in your publication or presentation indicating that the chart image is open and providing a link to the data and methods.

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