What is an Open Set & Open Interval? – Topology

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What is an Open Set?

 

In most of the text that you go through you will see that topology is defined in terms of open sets. You can think of an open set, as a set that does not have any boundary or boundary elements, whereas a closed set is one that has. First, we need to understand the relation between topology and open sets.

In topology, the generalization, which is an open interval for real numbers is an open set for topology.

When we work with spaces, we work with something called nearness. Say there are two points:

 

Figure 13 - Distance between two points
Figure 13 – Distance between two points

 

In metric space we define it by a function $d(x,y)$. Instead of defining a distance, we define it as open sets. $x∈y$ The set of all open sets on a space $X$ is called the topology on $X$. We can think of near each other as if there are a lot of open sets that contain both points. The other way round, if there are two points, that are never contained in the same open set – then they are very far apart.

So, in topology, we do not use things like metrics, distance, and equations. So if $ d(x,y)$ is small then $x$,$y$ are near.

 

 

Figure 14 - Open set & Closed set
Figure 14 – Open set & Closed set

 

Figure 14 shows that an open set is one, which does not have any boundary. lies within the disk but not outside is an open set But  which lies at the border is a closed set Figure 15

 

 

Figure 15 - Closed set & open set equations
Figure 15 – Closed set & open set equations

 

Inference thus can be drawn that open set is:

  • A set that does not have any border
  • Moving one point in one direction such that it is also in the set
  • Satisfies $x^{2}+y^{2}<r^{2}$
  • The open set is a generalization of a metric. Instead of using the distance function $d(x,y)$ we use open sets.
  • An open subset $\mathbb{R}$ of a subset of $E$ of $\mathbb{R}$ such that for every $x$ in $E$ there exists $\varepsilon>0$ such that $B_{\varepsilon }(x)$ is contained in $E$
  • Both $\mathbb{R}$ and the empty set are open

 

The reverse is also true. The complement of a subset $E$ of $\mathbb{R}$ is the set of all points in $\mathbb{R}$ which are not true in $E$.

So, in that case $[2,5)$ is not an open set but its complement  $(-\infty ,2)\cup (5,\infty )$ is open.

Hence, open set is an arbitrary metric space:

$B(x,r){:=y|y\in X,d(x,y)<r}$

 


What is an Open Interval?

 

An interval that does not include endpoints is called an open interval. It is denoted as $(a,b)$

 

Figure 16 - Intervals
Figure 16 – Intervals

Figure 16 gives the general notation. A filled rounded figure denotes not including end values while the other one shows including the end values.

It can be written as $0<x<20$ means all the numbers between $0$ and $20$ but not $0$ or $20$.

In order to know more about open intervals, you can consult any book. it gives you a basic idea and this is not at all complicated.

Next | Who was Pavel Sergeyevich Alexandrov?

 

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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