General Topology[Discrete and Indiscrete Topology With Examples]

Discrete and the Indiscrete Topology

Today i will be giving a tutorial on the discrete and indiscrete topology, this tutorial is for MAT404(General Topology), Now in my last discussion on topology, i talked about the topology in general and also gave some examples, in case you missed the tutorial click here to be redirect back.

Now we move onto the definition of a discrete topology. What is a discrete topology? It is defined as follow:

Let $X$ be a non-empty set and let $\mathcal{T}$ be the collection of all subset of $X$ i.e $\mathcal{T}$ is the power set of $X$. Then we say that $\mathcal{T}$ is the discrete topology on $X$ and the pair $(X,\mathcal{T})$ is called a discrete topological space.

Examples

If $X=\{1,2,3\}$ construct or define a discrete topology on $X$.

Solution

Since $X=\{1,2,3\}$ then all we need to do in order to construct a discrete topology on $X$ is to generate a power set on $X$. And we know, the formula for the power set is $P(X)=2^n$ where $n$=cardinality or number of elements in the set.

Now since $X$ consists of three elements, then the power is $2^3=8$ and it will consist of $8$ elements. that is pretty much easy i guess!!

Now $P(X)=\{\phi,X,\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\}\}$

Now we make $P(X)=\mathcal{T}$, hence $\mathcal{T}=\{\phi,X,\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\}\}$. hence since $\mathcal{T}$ is a collection of all subset of $X$ then we call $\mathcal{T}$ a discrete topology and the pair $(X,\mathcal{T})$ is called a discrete topology space.

Indiscrete Topology

Now we move on to defining what the induscrete topology is and also give some examples.

The indsicrete topology is defined as follows:

Let $X$ be a non-empty set and let $\mathcal{T}$ be the collection of the empty set($\phi$) and the set X. i.e $\mathcal{T}=\{\phi,X\}$, if $\mathcal{T}$ is a topology on $X$, then such a topology is called an indiscrete topology and the pair $(X,\mathcal{T})$ is called an indiscrete topological space.

Example

If $X=\{1,2,3\}$, construct or define an indiscrete topology on X.

Solution

Since $X=\{1,2,3\}$ if $\mathcal{T}=\{\phi,X\}$ then $\mathcal{T}$ is an indiscrete topology on $X$, if $\mathcal{T}$ consists of any other elements apart from $\phi$ and $X$,

then it will not be an indiscrete topology.

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