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A simple closed curve is a continuous loop in a plane that does not intersect itself, effectively dividing the plane into an interior and exterior region. It is a fundamental concept in topology and geometry, serving as the basis for more complex shapes and theorems, such as the Jordan Curve Theorem.
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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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Geometry is a branch of mathematics concerned with the properties and relationships of points, lines, surfaces, and shapes in space. It encompasses various subfields that explore dimensions, transformations, and theorems to understand and solve spatial problems.
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The Jordan Curve Theorem states that a simple closed curve in the plane divides the plane into an interior and an exterior region, with the curve itself being the boundary of both. This theorem is fundamental in topology as it establishes the basic property of continuous curves, highlighting the distinction between inside and outside in a two-dimensional space.
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A planar graph is a graph that can be embedded in the plane such that no edges intersect except at their endpoints. This property is crucial in fields like topology and graph theory, where it aids in solving problems related to map coloring and network design.
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Boundaries are limits or edges that define the scope of an entity, distinguishing what is included from what is excluded. They are essential in various fields to maintain order, structure, and clarity, whether in physical spaces, social interactions, or conceptual frameworks.
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Homeomorphism is a continuous bijective function between topological spaces that has a continuous inverse, preserving the topological properties of the spaces. It is a fundamental concept in topology, used to classify spaces by their intrinsic geometric properties rather than their extrinsic shape or form.
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Connectedness refers to the state of being linked or associated with others, fostering a sense of belonging and shared identity. It is fundamental to social cohesion and personal well-being, influencing how individuals interact within communities and networks.
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Compactness in mathematics, particularly in topology, refers to a property of a space where every open cover has a finite subcover, which intuitively means the Space is 'small' or 'bounded' in a certain sense. This concept is crucial in analysis and topology as it extends the notion of closed and bounded subsets in Euclidean spaces to more abstract spaces, facilitating various convergence and continuity results.
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Orientation refers to the process of aligning or positioning oneself or an object in relation to a specific direction or reference point. It is crucial in various fields, including navigation, psychology, and organizational behavior, as it helps individuals and systems effectively adapt and function within their environments.
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A curve is a continuous and smooth flowing line without any sharp turns or angles, often representing a mathematical function or path in geometry and calculus. It can be described in various forms such as parametric, implicit, or explicit equations, and is fundamental in understanding the behavior of functions and shapes in both two and three dimensions.
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Green's theorem provides a relationship between a line integral around a simple, closed curve C and a double integral over the plane region D bounded by C. It is a fundamental result in vector calculus that facilitates the conversion of a line integral into a more manageable double integral, often simplifying the computation of circulation and flux in a vector field.
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A closed curve is a continuous loop in a plane that starts and ends at the same point without crossing itself. It is a fundamental concept in topology and geometry, often used to describe boundaries and shapes, such as circles and ellipses.
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A Jordan curve is a continuous, non-self-intersecting loop in a plane, which, according to the Jordan Curve Theorem, divides the plane into an 'inside' and 'outside' region. This fundamental concept in topology is crucial for understanding the properties of simple closed curves and their role in complex analysis and geometry.
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A Dehn twist is a fundamental operation in the study of surface topology, specifically used to modify surfaces by cutting along a simple closed curve, twisting one of the resulting sides by 360 degrees, and re-gluing. This operation plays a crucial role in understanding the mapping class group of a surface, which is central to the classification of surfaces and the study of 3-manifolds.
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