Connectedness. 4 lemmas

1. The edge of a connected space

Let \( X \) be a topological space. Let \( A \) and \( B \) be subspaces of \( X \) with [some requirement].

If \( A \) is connected, then so is \( B \).

2. A [something] of a connected space is connected

Let \( f : X \to Y \) be a continuous map of topological spaces. If [something] then [something].

3. The [something] of two connected spaces is connected.

4. A space that has [a particular way of being composed] is connected.

Let \( X \) be a nonempty topological space and \( (A_i)_{i \in I} \) a family of subspaces covering \( X \). Suppose that \( A_i \) is [something] for each \( i \in I \) and that \( A_i \cap A_j \neq \emptyset \) for each \( i, j \in I \), then \( X \) is connected.