Seifert–van Kampen theorem

The Seifert–van Kampen theorem is a tool in topology that helps determine the fundamental group (which captures the basic shape features) of a space formed by combining simpler pieces. If a space is built from overlapping parts, and we understand each part's fundamental group, the theorem allows us to find the overall fundamental group by combining this information, much like assembling a puzzle. It’s especially useful for understanding the shape’s loops and holes, providing a systematic way to analyze complex structures from simpler, overlapping components.