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.(adsbygoogle = window.adsbygoogle || []).push({});

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

And here is how connectedness 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 ahomeomorphismand M, N are homeomorphic.

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: Let M be a metric space. If M has a proper clopen subset A, M isdisconnected. For there is aseparationof M into proper, disjoint clopen subsets. M isconnectedif it is not disconnected - it contains no proper clopen subset.

ExampleThe union of two disjoint closed intervals is not homeomorphic to a single interval. One set is disconnected and the other is connected.ExampleThe closed interval [a,b] is not homeomorphic to the circle S^{1}. 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.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.ExampleThe 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

Many thanks to anyone who can help me out.

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# Relationship b/w Connectedness and Homeomorphisms

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