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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.

http://en.wikipedia.org/wiki/Open_set#Motivation

&

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 $\tau_x$?

Thanks!

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