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Looking around trying to find questions concerning the intuition behind discrete/indiscrete topologies, I haven't found much towards the essence of what these particular topologies imply about the space (and why it is meaningful).

I realize that in a discrete topology, every set is both closed and open, as is the case with the indiscrete topology, however that the topologies are disconnected and connected respectively. That this somehow tells us about distinguishability between points; in the case of the discrete topology we have that every set of points can be at least disconnected from other sets of points, and that in the indiscrete topology, this is not the case.

Now the difference between the neighborhoods then would be then that for the Indiscrete topology, all the neighborhoods include all other points, but that in the discrete topology each point has a neighborhood which doesn't include the other points.

Am I correct so far? But I know there is much more, could someone expound on the intuition what is going on, maybe even including things such as levels of "seperated-ness", and perhaps what happens in between the Indiscrete and Discrete Topologies?

Thanks,

Brian

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A Topology $\tau$ on a set $X$ (a topological space (X,$\tau$) ) is a collection of subsets of $X$ such that the empty set, $X$ , the union of any subcollection and the intersection of any finite subcollection are in $\tau$.

Okay so we are know equipped with the definition of a topology. So we start thinking what kind of topologies I can give to a set. So lets look at the definition again. It says union and intersection of any sub collection is in $\tau$ . Naturally one candidate for this will be all subsets of a set $X$. You can see that this satisfies all the conditions for a topology and hence we have successfully defined a topology. The intuition behind this is that we first want to see that how does the set $X$, its subsets are in accordance with the definition of a topology on a set .

Also when we define $\sigma$- algebra on a set a successful candidate is again the set of all subsets of that set.

Indiscrete topology: Now in the indiscrete topology we consider the topology given by $\{X,\emptyset\}$. Now here again this is the most natural thing to think of because you will see that we want something such that the union of a subcollection is in the topology . Here we think of the set itself. Now you can see that the issue that occurs here is we don't have $\emptyset$ in it . So we have to include that to.

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