Open, closed and clopen sets
Given any nonempty set, we can determine if the elements of the sets are open, closed, clopen, not open, or not closed. This aspects is really confusing, no doubt.
This sets have wide range of applications especially in the study of topology. I will give a theorem and a simple proof of a topology and determine the nature of the elements of the sets.
In one of my previous lecture on topology, i discussed about discrete and indiscrete topology.
Before i proceed to giving the theorem and proof, i will give some basic definitions of the open, close and clopen sets in a topological space to refresh our memory.
Open sets
In any given topological space $(X,\mathcal{T})$, we say that the members of $\mathcal{T}$ are open sets.
Now this is really cool, there are some things we must know about this open sets, if we say a set is "not open" it does not imply closed, and if we say a set is "not closed" then we most know it does not imply open.
A very perfect example of the open sets are the sets $X$ and $\phi$ which are elements of $\mathcal{T}$, all other sets of $\mathcal{T}$ are also open sets.
Closed sets
The closed set is simply taking the complements of sets of $\mathcal{T}$, the closed set set is defined as "Let $(X,\mathcal{T})$ be a topological space and let $A\subset{X}$, then $A$ is called closed if and only if $A^c$ is open".
The sets $\phi$ and $X$ are closed sets in $\mathcal{T}$.
Clopen
The clopen set is a set that is both open and closed. The set $X$ and $\phi$ is a clopen set in any topological space.
In a discrete space all subsets of $X$ are clopen.
In an indiscrete space the only clopen subsets are $X$ and $\phi$.
Theorem: Let $X=\{a,b,c,d,e,f\}$ and $\mathcal{T}=\{X,\phi,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e,f\}\}$, show that the sets $\mathcal{T}$ contains the following:
(1)open,
(2)closed
(3)and clopen, subsets of $X$.
Proof: By definition of open sets we know that the members of $\mathcal{T}$ are open. This proves our statement $(1)$.
Now we need to show that $\mathcal{T}$ contains closed sets, to do this, we take the complements of the sets.
$X^c=X-X=\phi,\ \textit{hence}\ \phi \ \text{is closed}$
$\phi^c=X-\phi=X,\ X\ \text{is closed}$
$\{a\}=X-\{a\}^c=\{b,c,d,e,f\},\ \{b,c,d,e,f\}\ \text{is closed}$
$\{c,d\}^c=X-\{c,d\}=\{a,b,e,f\},\ \{a,b,e,f\}\ \text{is closed}$
$\{a,c,d\}^c=X-\{a,c,d\}=\{b,e,f\},\ \{b,e,f\}\ \text{is closed}$
$\{b,c,d,e,f\}^c=X-\{b,c,d,e,f\}=\{a\},\ \text{hence}\ \{a\} \textit{is closed}$.
Now we see that the sets $\phi,X,\{a\},\{b,c,d,e,f\}$ are all elements of $\mathcal{T}$, and thus we say they are open but they are also contained in our compliment subsets, and thus they are also closed sets. Now since it is evident that they are both open and closed, we conclude that they are clopen sets which proves case $(2)$ and $(3)$.
But if we take a good look at our compliments set, set $X$ and $\mathcal{T}$, we will also derive other sets that either closed but not open or otherwise.
Here we see that the set $\{b,c\}$ is neither open nor closed because it is not a subset of any set.
The set $\{c,d\}$ is open but not closed.
And finally, the set $\{a,b,e,f\}$ is closed but not open.
Hence we have thus show that the subsets of $X$ contains open, closed, clopen sets and other sets that are not open, closed nor clopen sets.
And that brings us to the end of the proof.
Given any nonempty set, we can determine if the elements of the sets are open, closed, clopen, not open, or not closed. This aspects is really confusing, no doubt.
This sets have wide range of applications especially in the study of topology. I will give a theorem and a simple proof of a topology and determine the nature of the elements of the sets.
In one of my previous lecture on topology, i discussed about discrete and indiscrete topology.
Before i proceed to giving the theorem and proof, i will give some basic definitions of the open, close and clopen sets in a topological space to refresh our memory.
 |
| Photo credit: https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Mathematical_Spaces.svg/220px-Mathematical_Spaces.svg.png |
Open sets
In any given topological space $(X,\mathcal{T})$, we say that the members of $\mathcal{T}$ are open sets.
Now this is really cool, there are some things we must know about this open sets, if we say a set is "not open" it does not imply closed, and if we say a set is "not closed" then we most know it does not imply open.
A very perfect example of the open sets are the sets $X$ and $\phi$ which are elements of $\mathcal{T}$, all other sets of $\mathcal{T}$ are also open sets.
Closed sets
The closed set is simply taking the complements of sets of $\mathcal{T}$, the closed set set is defined as "Let $(X,\mathcal{T})$ be a topological space and let $A\subset{X}$, then $A$ is called closed if and only if $A^c$ is open".
The sets $\phi$ and $X$ are closed sets in $\mathcal{T}$.
Clopen
The clopen set is a set that is both open and closed. The set $X$ and $\phi$ is a clopen set in any topological space.
In a discrete space all subsets of $X$ are clopen.
In an indiscrete space the only clopen subsets are $X$ and $\phi$.
Theorem: Let $X=\{a,b,c,d,e,f\}$ and $\mathcal{T}=\{X,\phi,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e,f\}\}$, show that the sets $\mathcal{T}$ contains the following:
(1)open,
(2)closed
(3)and clopen, subsets of $X$.
Proof: By definition of open sets we know that the members of $\mathcal{T}$ are open. This proves our statement $(1)$.
Now we need to show that $\mathcal{T}$ contains closed sets, to do this, we take the complements of the sets.
$X^c=X-X=\phi,\ \textit{hence}\ \phi \ \text{is closed}$
$\phi^c=X-\phi=X,\ X\ \text{is closed}$
$\{a\}=X-\{a\}^c=\{b,c,d,e,f\},\ \{b,c,d,e,f\}\ \text{is closed}$
$\{c,d\}^c=X-\{c,d\}=\{a,b,e,f\},\ \{a,b,e,f\}\ \text{is closed}$
$\{a,c,d\}^c=X-\{a,c,d\}=\{b,e,f\},\ \{b,e,f\}\ \text{is closed}$
$\{b,c,d,e,f\}^c=X-\{b,c,d,e,f\}=\{a\},\ \text{hence}\ \{a\} \textit{is closed}$.
Now we see that the sets $\phi,X,\{a\},\{b,c,d,e,f\}$ are all elements of $\mathcal{T}$, and thus we say they are open but they are also contained in our compliment subsets, and thus they are also closed sets. Now since it is evident that they are both open and closed, we conclude that they are clopen sets which proves case $(2)$ and $(3)$.
But if we take a good look at our compliments set, set $X$ and $\mathcal{T}$, we will also derive other sets that either closed but not open or otherwise.
Here we see that the set $\{b,c\}$ is neither open nor closed because it is not a subset of any set.
The set $\{c,d\}$ is open but not closed.
And finally, the set $\{a,b,e,f\}$ is closed but not open.
Hence we have thus show that the subsets of $X$ contains open, closed, clopen sets and other sets that are not open, closed nor clopen sets.
And that brings us to the end of the proof.
Posts
0 Comments
Comments