On real functions with graphs either connected or locally connected

Abstract

Let \( \mathcal{G} \) denote the family of all subspaces \( G \) of the plane \( \mathbb{R}^2\) such that \( G \) is the graph of a function from \( \mathbb{R}\) to \( \mathbb{R}\). We prove that \( \mathcal{G} \) has two subfamilies \( \mathcal{G}_1,\mathcal{G}_2 \) of connected spaces such that the cardinality of \( \mathcal{G}_1 \) is \( \mathbf{c} :=2^{\aleph_0} \) and the cardinality of \( \mathcal{G}_2 \) is \( 2^ \mathbf{c} \), every space in \( \mathcal{G}_1 \) is completely metrizable, each \( G\in\mathcal{G}_2 \) is a dense subset of \( \mathbb{R}^2\), and if \( X_1,X_2 \in \mathcal{G}_1\cup\mathcal{G}_2 \) are distinct then the space \( X_1 \) is neither homeomorphic to a subspace of \( X_2 \) nor homeomorphic to a proper subspace of \( X_1\). On the other hand, the family \( \mathcal{G} \) contains precisely \(\aleph_0 \) locally connected spaces up to homeomorphism, and if \( X\), \(Y \) are such spaces (including the case \(X=Y\)) then \(X \) is homeomorphic to some proper subspace of \( Y\). Furthermore, if \( \tau \) is a topology on the set \( \mathbb{R} \) finer than the Euclidean topology and the space \( ( \mathbb{R},\tau) \) is separable and locally connected, then the space is locally compact and homeomorphic to some space in \( \mathcal{G} \). In a very natural way we establish a complete classification of all these refinements \( \tau \) of the real line.