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mjkato
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Got it, thank you
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d1 and d2 aren't metric spaces - they are metrics.mjkato said:Homework Statement
Let d1(x,y) = |x1 − y1| + |x2 − y2| and d2(x,y) = ((x1 − y1)^2+(x2 − y2)^2)^(1/2) be metric spaces on R^2.
How do you describe a neighborhood?mjkato said:Prove that if a point (x1,x2) is a limit point of a subset under one metric, it is a limit point in the other.
Homework Equations
The Attempt at a Solution
I'm mostly lost here- I know a limit point contains atleast one point from the set in every neighborhood, but can't figure out how to translate that into a proof of this.
Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations, such as stretching or bending. It is concerned with the concept of continuity and the relationships between points, sets, and their neighborhoods.
A metric space is a mathematical concept that describes a set of points, where the distance between any two points is defined by a function called a metric. This function assigns a numerical value to the distance between two points, and it should satisfy certain properties such as non-negativity, symmetry, and the triangle inequality.
In R^2, metric spaces can have different properties depending on the metric used. For example, the Euclidean metric, which is based on the Pythagorean theorem, is commonly used in R^2. However, other metrics such as the Manhattan metric, which measures distance by adding the absolute differences between coordinates, can also be used in R^2.
Topology helps to define the structure of a metric space by studying the relationships between points, sets, and their neighborhoods. It provides a framework for understanding the behavior of functions and transformations in metric spaces, and it plays a crucial role in many areas of mathematics and science.
Basic topology is used in various fields such as physics, engineering, and computer science to model and analyze real-world phenomena. For example, topology is used in geographic information systems to analyze spatial data, in computer graphics to create 3D models, and in network theory to study communication and transportation networks.