A closed surface is a two-dimensional manifold that is compact and without boundary, meaning it completely encloses a volume in three-dimensional space. Examples include spheres and tori, which are essential in topology for understanding the properties of three-dimensional spaces.

Closed Surface

closed surface

two-dimensional manifold

compact

without boundary

encloses a volume

Three-dimensional space is a geometric setting in which three values, often referred to as dimensions, are required to determine the position of an element. It is the physical universe we live in, where objects have length, width, and height, allowing for the representation and manipulation of objects in a realistic manner.

three-dimensional space

A sphere is a perfectly symmetrical three-dimensional geometric object where every point on its surface is equidistant from its center, making it a fundamental shape in mathematics and physics. Spheres are prevalent in natural and artificial contexts, from celestial bodies to everyday objects, and their properties are crucial in fields such as geometry, calculus, and topology.

spheres

tori

Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations such as stretching and bending, but not tearing or gluing. It provides a foundational framework for understanding concepts of convergence, continuity, and compactness in various mathematical contexts.

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