Every path-connected metric space is connected

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• Sep 12th 2011, 07:48 AM
catnip
Every path-connected metric space is connected
Hi,

in my Calculus textbook there's a proof, that every path-connected metric space is connected, unfortunately, this proof makes use of some theorems of topology.

I don't know much about topology and have problems to find proofs for:

1. On every metric space the metric induces a topology.
2. A map is continuous regarding the metric iff it is continuous regarding the topology which the metric induces.
3. The image of a connected set is connected if the map is continuous.
4. $[0,1]$ is connected.

If you have links to these proofs or know a book where they are all explained in a simple manner, please let me know. Thanks a lot.
• Sep 12th 2011, 08:07 AM
Plato
Re: Every path-connected metric space is connected
Quote:

Originally Posted by catnip
in my Calculus textbook there's a proof, that every path-connected metric space is connected, unfortunately, this proof makes use of some theorems of topology.

What textbook are you using?
Quote:

Originally Posted by catnip
1. On every metric space the metric induces a topology.
2. A map is continuous regarding the metric iff it is continuous regarding the topology which the metric induces.
3. The image of a connected set is connected if the map is continuous.
4. $[0,1]$ is connected.

Almost any topology text will discuses all four of those concepts, albeit not in any one place in the text.
The first two are trivial once you have mastered the definition of a topology.

Robert B. Ash has a nice small textbook Real Variables with Basic Metric Space Topology. That should be sufficient to answer your questions. It is in a Dover edition.
• Sep 13th 2011, 07:31 AM
catnip
Re: Every path-connected metric space is connected
Thanks for the hint, I was looking for a book like this.

Of course you're right, it's discussed in nearly topology books, but they are mostly MUCH to broad and abstract for my needs.

And actually I'm not using a textbook... it's a preliminary draft of a textbook from my prof.