topology of Euclidean spaces

Topology in Euclidean spaces concerns properties related to the notions of closeness and continuity that remain unchanged under continuous transformations. Think of it as studying the shape of objects without focusing on exact sizes or distances—only how they are connected and arranged. For example, a coffee mug and a doughnut are topologically equivalent because each can be deformed into the other without tearing or gluing. In Euclidean spaces (like familiar 2D and 3D spaces), topology helps us understand concepts like open and closed sets, neighborhoods, and boundaries, providing a framework to analyze continuity, limits, and spatial relationships abstractly and flexibly.