Prove that if {Aα}α∈A is a collection of path connected...

Prove that if $\left\{A_\alpha\right\}_{\alpha \in A}$ is a collection of path connected subsets of a topological space $X$, and $\bigcap_{\alpha \in A} A_\alpha$ is nonempty, then $\bigcup_{\alpha \in A} A_\alpha$ is path connected.

Key Concepts

Path Construction via Concatenation

Path construction via concatenation involves joining two or more continuous paths at common points to form a new continuous path. This method is particularly useful in proofs where individual subsets are path connected and share a common point. By concatenating the paths within each subset through the common intersection, one can demonstrate that the entire union is path connected.

Union and Intersection of Sets

The union and intersection of sets are basic operations in set theory that have significant implications in topology. The intersection of sets being nonempty is crucial because it provides a common point through which different path connected subsets can be joined. This common intersection is the key to constructing a continuous path across the union of the subsets.

Path Connectedness

Path connectedness is a property of a topological space where any two points can be joined by a continuous path within the space. This concept is fundamental in topology as it provides a way to understand how points in a space relate through continuous curves, rather than through more abstract connectedness properties.

Continuous Functions

Continuous functions are mappings between topological spaces that preserve the structure of neighborhoods. In the context of path connectedness, the continuous image of a closed interval (commonly used to represent a path) is a way to construct paths between points, which is essential when proving the connectivity properties of unions of subsets.

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