What is Homeomorphism Type? Definition & Examples

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Homeomorphism type refers to the classification of topological spaces based on their homeomorphisms, meaning two spaces share the same homeomorphism type if they can be transformed into one another through continuous deformations. This concept is crucial in distinguishing between spaces that may have similar properties, such as homotopy type, but differ in their topological structure. For instance, lens spaces can exhibit the same homotopy type while maintaining distinct homeomorphism types. The classification of 3-dimensional manifolds up to homeomorphism type indicates that these manifolds can be categorized based on their topological equivalence. Understanding homeomorphism type is essential for deeper insights into the properties and relationships of various mathematical objects.
iLoveTopology
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I have one other question and I'd appreciate any insight in to. What exactly is "homeomorphism type"? I understand well what a homeomorphism is, but not what a homeomorphism type is. For example, I read about lens spaces and read things like "some lens spaces have the same homotopy type but not the same homeomorphism type". Or I've read "3-dimensional manifolds can be classified up to homeomorphism type". What exactly does this mean? Is this something like the set of all possible homeomorphisms on a mathematical object? Homeomorphisms to what? When I look online and in my textbook I can't seem to find a definition for "homeomorphism type"

Thank you very much!
 
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M and N have the same homeomorhpism type iff they are homeomorphic.
 

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