# Basic topology (differing metric spaces in R^2)

• mjkato
In summary, the conversation discusses the concept of metric spaces and their relationship to subsets and limit points. It is stated that d1 and d2 are metrics, not metric spaces. The group (R2, d1) and (R2, d2) are examples of metric spaces, where a metric space is defined as a set with a metric or measure. The task at hand is to prove that if a point (x1, x2) is a limit point of a subset under one metric, it is also a limit point under the other metric. The approach to this proof involves understanding the concept of a neighborhood and how it relates to limit points.
mjkato
Got it, thank you

Last edited:
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.
d1 and d2 aren't metric spaces - they are metrics.

(R2, d1) is a metric space, and so is (R2, d1). In other words, a metric space consists of a set (such as R2) together with a metric (or measure).
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.

## 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.
How do you describe a neighborhood?

## 1. What is topology?

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.

## 2. What is a metric space?

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.

## 3. How are metric spaces different in R^2?

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.

## 4. What is the role of topology in metric spaces?

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.

## 5. How is basic topology useful in real-world applications?

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.

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