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Taking the quotes

Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topological space there exists an open set not containing another (distinct) point, the two points are referred to as topologically distinguishable.


Your intuition about the "fuzziness" of open sets is correct.

Inuitively, if U is an open set in the topological space X and x∈U, then U necessarily contains any other point in X that is "near enough" to x

Motivation for the concept of "open set" in topology.

seriously for a scurrilous moment, how does one explain the following example:

$$(X = \{a,b,c,d,e,f\},\tau_x = \{\varnothing , X,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e,f\}\})$$

& non-example:

$$(Y = \{1,2,3,4,5\},\tau_y = \{\varnothing , X,\{1\},\{3,4\},\{1,3,5\},\{2,3,4\}\})$$

of an open-set topological space in terms of this intuition? Explicitly:

a) What does $\{c,d\}$ mean? It means $c$ is 'close' to $d$, right? However does it also mean that $c$ & $d$ are indistinguishable in our topology, since they are not in sets of their own?

b) What does the fact that $\{a\}$ & $\{c,d\}$ are in different sets really mean? I'm guessing it means that $a$ is distinguishable from $c$ & $d$ (which themselves are indistingishable) but that $a$ is also close to $c$ (& $d$) since $\{a,c,d\}$ is in our topology. We have to be careful here, because we could interpret $\{a,c,d\}$ as meaning that $a$ is indistinguishable from $c$ & $d$ as well by this logic, so does also including $a$ & $c$ (with $d$) in it's own set mean something like "$a$, $c$ & $d$ are indistinguishable, except for the fact that $a$ is distinguishable from $c$ & $d$"?

c) If we didn't include $\{a,c,d\}$, would it would mean that while $a$ is distinguishable from $c$ (& $d$) in our topology, we cannot determine whether or not $a$ is 'near' to $c$ & $d$, thus we cannot construct any kind of 'geometry' on this set? (In other words, we cannot compare $a$ to $c$ & $d$?). Again we have to be careful because $a$, $c$ & $d$ are in $X$ together, thus would not including $\{a,c,d\}$ mean something like "$a$, $c$ & $d$ are indistinguishable & close to each other (since they're in $X$!), except for the fact that $a$ is distinguishable from $c$ & $d$ (since $\{a\}$ & $\{c,d\}$ exist in $\tau_y$), but since $a$, $c$ & $d$ are not close to each other (since $\{a,c,d\}$ doesn't exist) we cannot compare $\{a\}$ to $\{c,d\}$ thus we don't have a topology.

This perspective basically means reading from bigger sets down to smaller sets with two interpretations, closeness & distinguishability, & using these as checks. Weak sauce? Even if it's weak, as long as it works it's great to me!

d) What does including the set $\{b,c,d,e,f\}$ achieve? I mean we could leave it out & still have a topology, thus I guess it's just a way of saying that every element other than $a$ is indistinguishable in this specific topology right?

e) Can we use these interpretations to explain why $\tau_y$ is not a topology in the way I've examined not including $\{a,c,d\}$ in $\tau_x$?


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