Homemorphism of two metric space

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SUMMARY

The discussion confirms that a bijective map between the open balls of two metric spaces indeed implies a homeomorphism, provided certain conditions are met. Specifically, if the map \( f: X \to Y \) is continuous on a basis for \( Y \) and acts as an open map on a basis for \( X \), then it qualifies as a homeomorphism. This conclusion is based on the properties of continuity and open mappings in topology.

PREREQUISITES
  • Understanding of bijective mappings in topology
  • Familiarity with metric spaces
  • Knowledge of continuity and open maps
  • Basic concepts of homeomorphisms
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  • Study the properties of bijective functions in topology
  • Learn about continuity in metric spaces
  • Explore the concept of open maps and their significance
  • Investigate examples of homeomorphisms in various metric spaces
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Mathematicians, students of topology, and anyone interested in the properties of metric spaces and homeomorphisms.

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If I create a bijective map between the open balls of two metric spaces, does that automatically imply that this map is a homemorphism?
 
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Yeah, it does (if I'm interpreting your statement correctly). You only need to check continuity on a basis, so if you have a bijection ##f: X \to Y## between two metric spaces and it is continuous on a basis for ##Y## and an open map on a basis for ## X## it's a homeomorphism.
 
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