Relationship b/w Connectedness and Homeomorphisms

In summary: Essentially, the concept of being connected or disconnected is preserved under homeomorphisms, meaning that if one space is connected and the other is not, they cannot be homeomorphic to each other. This is because homeomorphisms preserve open sets, and if a space is disconnected, it means that it can be divided into two open and disjoint sets, which cannot be preserved under a homeomorphism. Therefore, if one space is connected and the other is not, they cannot be homeomorphic.
  • #1
rekovu
1
0
I'm using Real Mathematical Analysis by Pugh to supplement my analysis class, and the book has been clear thus far, but I've been stuck for days on a concept I've had a hard time understanding.

Just for reference, here is how a homeomorphism is defined:

Let M and N be metric spaces. If f: M->N is a bijection and f is continuous and the in verse bijection f-1:N->M is also continuous is also continuous then f is a homeomorphism and M, N are homeomorphic.

And here is how connectedness is defined:

Let M be a metric space. If M has a proper clopen subset A, M is disconnected. For there is a separation of M into proper, disjoint clopen subsets. M is connected if it is not disconnected - it contains no proper clopen subset.

I understand that M connected and M homeomorphic to N implies N connected, and that M connected, f:M->N continuous, and f onto implies N connected. However, what I don't understand are examples such as the following:

Example The union of two disjoint closed intervals is not homeomorphic to a single interval. One set is disconnected and the other is connected.

Example The closed interval [a,b] is not homeomorphic to the circle S1. for removal of a point x in (a,b) disconnects [a,b] while the circle remains connected upon removal of any point. More precisely, suppose that h: [a,b] is a homeomorphism. Choose a point x in (a,b), and consider X = [a,b] \ {x}. The restriction of h to X is a homeomorphism from X onto Y, where Y is the circle with one point, hx, removed. But X is disconnected, while Y is connected. Hence h can not exist and the segment is not homeomorphic to the circle.

Example The circle is not homeomorphic to the figure eight. Removing any two points of the circle disconnects it, but this is not true of the figure eight. Or, removing the crossing point disconnects the figure eight, but removing no points disconnects the circle. pg 85 Pugh

What I don't understand is how M connected and N disconnected implies M, N not homeomorphic, as in the first example. If I take this as being true, I understand the logic of the second example, with the detailed explanation of the process of removing a point. However, I still don't see M connected and N disconnected implying M,N not homeomorphic.

Many thanks to anyone who can help me out.
 
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  • #2
rekovu said:
I'm using Real Mathematical Analysis by Pugh to supplement my analysis class, and the book has been clear thus far, but I've been stuck for days on a concept I've had a hard time understanding.

Just for reference, here is how a homeomorphism is defined:



And here is how connectedness is defined:



I understand that M connected and M homeomorphic to N implies N connected, and that M connected, f:M->N continuous, and f onto implies N connected. However, what I don't understand are examples such as the following:







What I don't understand is how M connected and N disconnected implies M, N not homeomorphic, as in the first example. If I take this as being true, I understand the logic of the second example, with the detailed explanation of the process of removing a point. However, I still don't see M connected and N disconnected implying M,N not homeomorphic.

Many thanks to anyone who can help me out.

homeomorphisms preserve open sets.
 
  • #3
rekovu said:
I understand that M connected and M homeomorphic to N implies N connected, and that M connected, f:M->N continuous, and f onto implies N connected. However, what I don't understand are examples such as the following:

So if N is not connected, and a homeomorphism exists from M to N when M is connected, we get a contradiction (because N is connected and not connected at the same time)
 
  • #4
rekovu said:
I understand that M connected and M homeomorphic to N implies N connected...

rekovu said:
What I don't understand is how M connected and N disconnected implies M, N not homeomorphic...

Your understanding of one should imply your understanding of the other!
 
  • #5


The concept of connectedness and homeomorphisms are closely related because they both deal with the topological properties of a space. In particular, a homeomorphism preserves the topological structure of a space, meaning that two spaces that are homeomorphic are essentially the same in terms of their topological properties.

In the first example, the union of two disjoint closed intervals is not homeomorphic to a single interval because the union is disconnected while the single interval is connected. This is because a homeomorphism preserves the topological structure, and in this case, the topological structure of the two spaces is different. The union of two disjoint closed intervals can be separated into two proper, disjoint clopen subsets, while a single interval cannot be separated in this way. Therefore, the two spaces are not homeomorphic.

In the second example, the closed interval [a,b] and the circle S1 are not homeomorphic because removing a point from [a,b] disconnects the space, while removing a point from S1 does not disconnect the space. This again shows that the topological structures of the two spaces are different, and therefore, they cannot be homeomorphic.

In the third example, the circle and the figure eight are not homeomorphic because removing any two points from the circle disconnects it, while removing two points from the figure eight does not disconnect it. This again shows that the topological structures of the two spaces are different, and therefore, they cannot be homeomorphic.

In summary, the relationship between connectedness and homeomorphisms is that a homeomorphism preserves the topological structure of a space, and if two spaces have different topological structures, they cannot be homeomorphic. This is why M connected and N disconnected implies M, N not homeomorphic.
 

1. What is the definition of connectedness in relation to homeomorphisms?

Connectedness refers to the topological property of a space that cannot be separated into two disjoint open subsets. In other words, a connected space is a single, unbroken space that cannot be divided into smaller pieces without breaking its continuity.

2. How are homeomorphisms related to connectedness?

Homeomorphisms are continuous bijective maps between two topological spaces. In the context of connectedness, homeomorphisms preserve the connectedness of a space. This means that if two spaces are homeomorphic, they have the same connectedness properties.

3. Can a homeomorphism change the connectedness of a space?

No, a homeomorphism cannot change the connectedness of a space. As mentioned before, homeomorphisms preserve connectedness. This means that if a space is connected, its homeomorphic image will also be connected, and vice versa.

4. How does connectedness affect the behavior of homeomorphisms?

Connectedness has a significant impact on the behavior of homeomorphisms. In particular, it determines the number of different topological properties that can be preserved by a homeomorphism. For example, if a space is connected, any homeomorphism between that space and another must also preserve connectedness.

5. Are there any other topological properties that are related to both connectedness and homeomorphisms?

Yes, there are other topological properties that are related to both connectedness and homeomorphisms. Some examples include compactness, path-connectedness, and Hausdorffness. These properties are also preserved by homeomorphisms and can have an impact on the connectedness of a space.

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