Relationship b/w Connectedness and Homeomorphisms

[rekovu]

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.

 

[lavinia]

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.

 

[Office_Shredder]

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)

 

[Jamma]

I understand that M connected and M homeomorphic to N implies N connected...

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!