Definition 1

Let X and Y be topological spaces and let $f$ be a continuous mapping of X into Y. One says that the mapping $f$ is relatively connected if every point of Y possesses a fundamental system

of neighbourhoods consisting of sets V such that $\overset{-1}{f}(V)$ is connected and nonempty.

Let $f: X\rightarrow Y$ be a continuous mapping; in order that $f$ be relatively connected, it is necessary and sufficient that every point of Y possess a neighbourhood V such that the mapping $f_V:\overset{-1}{f}(V)\rightarrow V$ induced by $f$ be relatively connected.

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a relatively connected continuous mapping. The image of $f$ is dense in Y. For every open subset of Y, the mapping $f_V:\overset{-1}{f}(V)\rightarrow V$ is relatively connected.