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Homeomorphism

Given topological spaces \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\), a function \(f:X\to Y\) is called a 'homeomorphic' if \(f\) is bijective and continuous, and its inverse \(f^{-1}:Y\to X\) is also continuous. If such a map exists then one writes \(X\cong_\mathbf{Top}Y\) and calls \(X\) and \(Y\) 'homeomorphic', the topological word for isomorphic. One says that \(f:X\to Y\) is 'homeomorphic onto its image' if \(\tilde{f}:X\to \mathrm{im}(f)\) is a homeomorphism when \(\mathrm{im}(f)\) is endowed with the subspace topology.