# Path-connectedness. 4 lemmas

Path-connectedness implies connectedness, the converse is [true or not true?].

The topologist's sine curve is an example of a space that is [something]. Leinster covers the topologist's sine curve in some detail.

Below, we introduce an iff statement that does hold. It has the form: something ∧ connected ⟺ path-connected. After this, 3 conditions that each imply path-connectedness are presented.

#### Path-connected ⟺ connected and [something]

Let \( X \) be a topological space. \( X \) is path-connected if and only if \( X \) is connected and [...].

1. Corollary. Every connected [some type of Euclidean set] is path-connected.

#### 2. Path-connectedness and continuous functions

Let \( f : X \to Y \) be a continuous map of topological spaces. If \( X \) is path-connected, then [...].

#### 3. Path-connectedness and products

The product of two [path-connected...is there any other requirements?] spaces is path-connected.