homeomorphism as isomorphism in Top
1. Proposition
Let \((X,\mathcal{T})\) and \((Y,\mathcal{T}_Y)\) be topological spaces and \(f: X \rightarrow Y\) a continuous map
TFAE:
- \(f\) is a homeomorphism
- \(f\) is an isomorphism in category Top
2. Proof(sketch)
2.1. 1) \(\implies\) 2)
in category set
- furthermore, \(f^{-1}\) is a welldefined morphism, because \(f^{-1}\) is continuous
2.2. 2) \(\implies\) 1)
follows from \(f^{-1}\) as welldefined morphism