Why is the void open?

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In summary, the conversation discusses the axiom of a topology that the void is open and how it follows from the definition that the void is empty. The conversation also clarifies that the definition always applies, even to the empty set. Additionally, it is mentioned that the empty set is both open and closed.
  • #1
quasar987
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I'm not asking why it is an axiom of a topology that the void is open, but rather, when the topology of R^n is developped and open sets are defined as sets such that for any point in the set, we can find an epsilon-ball centered on that point that is entirely contained in the set.

My book says that it follows from the dfn that the void is empty. How is that? If we argue that "since the void has no point, then it is true that for all points, we can find and epsilon-ball, etc.", then the opposite is just as true: "Since there are no point, we can say that for all point, we can never find an epsilon-ball, etc."

There is no points in the void, so the definition simply does no apply it seems!Similar question: what's the boundary of the void? is it the void or the whole of R^n?
 
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  • #2
quasar987 said:
My book says that it follows from the dfn that the void is empty. How is that? If we argue that "since the void has no point, then it is true that for all points, we can find and epsilon-ball, etc.", then the opposite is just as true: "Since there are no point, we can say that for all point, we can never find an epsilon-ball, etc."

You're correct that both arguments are valid, but note that "Every point does not have X" is NOT the opposite of "Every point has X". In order for the null set not to be open, we would need that the contradiction of "Every point has X" hold, which is of course "there is a point which does not have X".

There is no points in the void, so the definition simply does no apply it seems!

The definition always applies. Either every point has a property, or one of the points doesn't.

Similar question: what's the boundary of the void? is it the void or the whole of R^n?

Well the null set is closed, so that ought to answer your question.
 
  • #3
Ok, I see! Thx DW
 
  • #4
No problem. It does sound like you might want to brush up more on your logic if you intend to study mathematics at this level.
 
  • #5
This has been my conclusion as well :smile:
 
  • #6
consider a statement of form "for all elements of set S, property P is true".

If S is empty this statement is true, :"vacuously".

this aNSWERS YOUR QUESTION, say in a metric space. i.e. openness is defined by a ":universal" quntifier: "for all p in S, there is an open ball around p also contained in S".
 
  • #7
mathwonk said:
consider a statement of form "for all elements of set S, property P is true".

If S is empty this statement is true, :"vacuously".

this aNSWERS YOUR QUESTION, say in a metric space. i.e. openness is defined by a ":universal" quntifier: "for all p in S, there is an open ball around p also contained in S".


S contains all its limit points.
If rephrased (i.e., as a universally quantified implication) and taken as definition of closed set in a metric space, we arrive at same vacuous truth (i.e, antecedent false) wrt. empty set.

Empty set both open and closed.
 
Last edited:

1. Why is the void open?

The void, also known as the cosmic void or dark void, is a large region of space that appears to have very few or no galaxies. This can be attributed to the expansion of the universe, which causes the space between galaxies to grow larger and create voids.

2. How big is the void?

The size of the void can vary, but it is typically on the scale of hundreds of millions of light years. This makes it one of the largest structures in the universe, but it is still relatively small compared to the overall size of the universe.

3. What causes the void to be open?

The void is open due to the gravitational pull of matter. As matter clumps together to form galaxies and clusters, it creates areas of high density and low density. The void is a result of low density regions where matter is not as concentrated.

4. Is the void completely empty?

No, the void is not completely empty. It still contains matter, such as gas and dust, but at much lower densities compared to the rest of the universe. This makes it difficult to detect and study, but it is not a completely empty space.

5. Is the void expanding?

Yes, the void is expanding along with the rest of the universe. This is due to the ongoing process of cosmic expansion, where the space between galaxies continues to grow. As the void expands, it will become even more empty as matter is spread out even further.

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