Similar functions and their properties (Q2843899)

The functions \(f\:X\to Y\) and \(g\: X\to Z\) (\(X\), \(Y\) and \(Z\) are topological spaces) are said to be similar (strongly similar) if the nets \(\bigl \{f(x_\gamma)\bigr \}_{\gamma \in \Gamma}\) and \(\bigl \{g(x_\gamma)\bigr \}_{\gamma \in \Gamma}\) are simultaneously convergent or divergent for each convergent net \(\{x_\gamma \}_{\gamma \in \Gamma}\) in \(X\) (for each net \(\{x_\gamma \}_{\gamma \in \Gamma}\) in \(X\)). Some properties of similar (strongly similar) functions are investigated.