Definition
3
[2B9]Let \((X, τ )\) and \((Y, σ)\) be two topological spaces, with \((Y, σ)\) Hausdorff; let \(f:X\to Y\) be a function.
It is said that \(f\) is continuous in \(x_ 0\) if \(\lim _{x→x_ 0} f (x) = f(x_ 0)\).
It is said that \(f\) is continuous if (equivalently)
if \(f\) is continuous at every point, that is \(\lim _{x→y} f (x) = f(y)\) for every \(y∈ X\), or
if \(f^{-1}(A)∈ τ\) for each \(A∈ σ\).
(Thm. 5.7.4 in the notes [ 3 ] .).
A continuous bijective function \(f:X\to Y\) such that the inverse function \(f^{-1}:Y\to X\) is again continuous, is called homeomorphism.