The discussion focuses on providing an example of a homeomorphism from a subset U to its image φ(U) where φ(U) is not open in the topological space M. The identity map I_d is utilized as a homeomorphism, demonstrating that when U is not open in M, the image I_d(U) retains this property under the subspace topology derived from M. This establishes a clear relationship between the properties of U and its image in the context of topology.
PREREQUISITESMathematicians, particularly those specializing in topology, students studying advanced mathematics, and anyone interested in the properties of homeomorphisms and their applications in topological spaces.
Take a subset ##U\subset M##, which is not open and the identity map.cianfa72 said:TL;DR Summary: An example of homeomorphism onto the image that isn't open in the target
Can you provide an example of homeomorphism onto the image φ:U→φ(U) where the image φ(U)⊂M is not open in M w.r.t. its assigned topology ?
Thanks.
Ah ok, you mean a not open ##U \subset M## topologized with the subspace topology from ##M## taken as domain of the identity map ##I_d## (homeomorphism onto the image ##I_d(U)## w.r.t. the subspace topology from ##M##).martinbn said:Take a subset ##U\subset M##, which is not open and the identity map.