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. 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.