Metric Topological Groups

A metric topological group is a mathematical structure where a set has both a group operation (like combining two elements) and a notion of distance between elements (called a metric). The key idea is that the group operations are compatible with this distance, meaning multiplying or inverting elements smoothly changes their positions without sudden jumps. This combination allows mathematicians to analyze symmetries and continuous transformations in spaces where you can measure how "close" or "far apart" elements are, with applications in areas like physics, analysis, and geometry.