Local path-connectedness

Local path-connectedness is a property of a space where, around every point, there exists a small neighborhood that is itself connected in a way that any two points within it can be joined by a continuous path lying entirely inside that neighborhood. In simpler terms, it means that very small regions near any point are "path-friendly"—you can smoothly travel from one point to another within that tiny area without leaving it. This concept helps us understand the structure and connectivity of spaces on a small scale, ensuring no isolated "islands" or complex barriers within these local neighborhoods.