What is Euclidean Space?

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Topology and Topological Space

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What is Euclidean Space?

 

Topological space is a generalization of Euclidean space. When we say this, we mean that topological spaces are generalizations of metric spaces which in turn is a generalization of Euclidean space.

As I have mentioned earlier the more we move towards generalization, the more things become easy. In this section, we will see from Euclidean space, how we move towards metric space and then to the topological space. 

 

Figure 11 - Euclidean space to topological space
Figure 11 – Euclidean space to topological space

 

Figure 11 gives you an idea of the way it moves.

 

Figure 12 - Measuring distance between points
Figure 12 – Measuring distance between points

 

The conventional way in which we measure the distance between two points is by a non-negative function $d(x,y)$.  We can further state $d:S\times S\rightarrow \mathbb{R}$ 

Here $S$ is a set and $d$ is a metric on $S$

A function $d:X\times X\rightarrow \mathbb{R}$ is a metric for any $x,y,x\in X$

  1. $d(x,y)=0$ iff $x=y$
  2. $d(x,y)=d(y,x)$
  3. $d(x,y)≤d(x,z)+d(z,y)$

We refer to $(x,D)$ as a metric space.

In order to explain Euclidean space, first, we need to understand what is a ‘tuple’. An n-tuple or simply tuple, is another word for the list, is an ordered set of n elements. You can call it a vector or n vector. The n-tuple is called points. Following are some of the examples of tuples:

1- monad
2 – pairs
3 – triple
4 – quadruple
5 – quintuple

We can also say:

$\mathbb{R}^{n}$ is a vector space

$\mathbb{R}^{2}$ is a Euclidean plane

$\mathbb{R}^{1}=\mathbb{R}^{n}$ a set of real numbers

Euclidean $n$ space is also called Cartesian space, a space of all tuples of real numbers say:

$x_{1},x_{2}…x_{n}$

So we can summarize Euclidean space as:

  • A fundamental space of classical geometry
  • It is a finite-dimensional inner product space over a real number
  • Line in Euclidean subspace is of dimension one
  • Two subspaces $S$ and  $T$ in a Euclidean space are parallel if they have the same direction
  • The distance between two points is: $d(p,q)=|pq|
  • Two non zero vectors $v$ and $u$ of $\varepsilon$ of  vector space are perpendicular if $u\cdot v=0$

Now a Euclidean distance makes a Euclidean space and thus a topological space. This topology is called Euclidean geometry. So, you see in Euclidean geometry we also have a topology.

 

Next | Euclidean Axioms to Topological Axioms

 

See Also | More Topology Articles


Further References:

  • Topology – James R. Munkres
  • Topology without tears – Sidney A.Morris

Article Advisor:

  • Richard Sot (PhD. Mathematics, University of Rochester, Rochester, NY)

Website references:

  1. https://mathworld.wolfram.com
  2. https://www.wikipedia.org/
  3. https://math.stackexchange.com/
  4. https://www.researchgate.net/publication/343635292

 

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