# Topology on Eculidean n-space(ℝ^n)

• davechrist36
In summary: To show that the intersection of any finite number of sets is also in the collection, take the intersection of all finite subcollections of sets in the collection. So, in summary, the topology for a set X is a collection of subsets of X, T, such that:1) X is in the collection.2) The empty set is in the collection.3) The union of any sets in the collection is also in the collection.4) The intersection of any finite number of sets in the collection is also in the collection.
davechrist36
Hey guys in fact here is my first time to have interaction over this forum!

I've already read how one can show the topology in ℝ(real Line) which is usual called standard topology fulfill the three condition fro to be topology. however,

I want to make inquiry on how can i proof whether the topology in ℝ^n(Euclidean n-space) satisfying the condition of topology, thus making topology ?

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...what? You want to prove that "the topology" on Rn is a topology?

I think that what davechrist37 is saying is that he wants to prove that the collection of what are called "open sets" as normally defined on Rn is a topology. Of course, those are defined by the metric $d(x,y)= \sqrt{(x_1-y_1)^2+ (x_2- y_2)^2+ \cdot\cdot\cdot+ (x_n- y_n)^2}$ so it is only really necessary to prove that that is a metric. I believe that is typically done in any introductory Analysis course.

thanx guys for you immediate concern. in fact i was looking for how the collection of open sets over Euclidean-n space forms topology; however right yesterday I got a material w/c help me how can I go thru the proof of that. let me glance ovr that and i'll reflect it here again.

let B^(n )∁ R^n denote the open unit ball in R^n with center at the orgine.i.e
B^(n )={x∈R^n:|x|<1}.Then how I can show or Prove the map f:R^n-→B^(n ) given by
f:x⟼x/(1+|x| ) ϵB^(n )
is well defined and gives a homeomorphism B^(n )≅ R^n

That last is a completely different question, and much harder, from your first question!
That, at least is relatively straight forward. Given a set X, a topology for X is a collection of subsets of X, T, such that:
1) X is in the collection.
2) The empty set is in the collection.
3) The union of any sets in the collection is also in the collection.
4) The intersection of any finite number of sets in the collection is also in the collection.

The "standard topology" for Rn is the collection of open sets where a set, A, is open if and only if "for ever x in A, there exist a number $\delta> 0$ such that the ball, $B_\delta(x)$, defined as $\{y | d(x,y)< \delta$ is a subset of A". Here d(x, y) is the "standard metric" on Rn: if $x= (x_1, x_2, ..., x_n)$, and $y= (y_1, y_2, ..., y_n)$, then $d(x,y)= \sqrt{(x_1- y_2)^2+ (x_2- y_2)^2+ \cdot\cdot\cdot+ (x_n- y_n)^2}$.

Now, suppose x is in $\cup \{U_i\}$ where $\{U_i\}$ is a collection of open sets A. Then there exist some specific $U_i$ containing x. Since $U_i$ is open, there exist $\delta> 0$ such that $B_\delta(x)\subset U_i$. But if every point of $B_\delta(x)$ is in $U_i$j it is certainly in the union so $B_\delta(x)\subset \cup \{U_i\}$ and so $\cup \{U_i\}$ is an open set.

Intersection is a little trickier and why we need "finite". Suppose $x\in \cap U_i$. Then $x\in U_i$ for all i. Since every $U_i$ is open, there exist $\delta_i> 0$ such that $B_{\delta_i}(x)\subset U_i$ for every i. Here's where we need "finite". Since the set of all such $\delta_i$ is finite, there exist a smallest $\delta_k$. Then $B_{\delta_k}(x)$ is a subset of all such $B_i(x)$ and so is in every $U_i$ and so in their intersection. $B_k(x)$ for that k is in $\cap \{U_i\}$. Since x could be any point in $\cap\{U_i\}$, $\cap \{U_i\}$ is open.

Now that we have shown that the union of any sub-collection is also in the collection, to show that the entire space, A, is in the collection, take the union of all sets in the collection. To show that the empty set is in the collection, take the union of of the empty subcollection.

## 1. What is topology on Euclidean n-space?

Topology on Euclidean n-space is a mathematical concept that studies the properties of space and the relationships between points, lines, and shapes in n-dimensional spaces. It is a branch of mathematics that deals with the study of topological spaces, which are sets of points with a defined structure and set of rules.

## 2. What are the key elements of topology on Euclidean n-space?

The key elements of topology on Euclidean n-space include the concept of open sets, continuity, and connectedness. Open sets are sets of points that do not include their boundary points, while continuity refers to the ability to continuously transform one set into another. Connectedness refers to the ability to move from one point to another without crossing any gaps or holes.

## 3. How is topology on Euclidean n-space different from other branches of mathematics?

Topology on Euclidean n-space is different from other branches of mathematics in that it focuses on the global properties of spaces instead of their local properties. It also allows for transformations that preserve the structure of the space, rather than just the shape or size of the objects within it.

## 4. What are some real-world applications of topology on Euclidean n-space?

Topology on Euclidean n-space has a wide range of applications in various fields, including physics, engineering, and computer science. It is used to study the properties of space and objects in different dimensions, such as in string theory and quantum mechanics. It is also used in computer graphics and image processing to analyze and manipulate complex shapes and structures.

## 5. What are some open questions and challenges in topology on Euclidean n-space?

One of the main challenges in topology on Euclidean n-space is the study of high-dimensional spaces, where traditional geometric intuition may not apply. Another open question is the classification of topological spaces in higher dimensions, which is still an active area of research. Additionally, the development of topological methods and techniques for solving real-world problems is an ongoing challenge for the field.

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