Compact Sets
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SOUP /SCI/ what was your definition of compactness? / define a compact set in your own words; feel free to define other terms as you please.
Example:
Definition: A set [math]E[/math] is [math]compact[/math] iff, for every family of sets [math]{G_α}_(α \in A)[/math] of *open* sets such that [math]E \subset \bigcap_(α \in A)G_α[/math], there is a finite set [math] {α_1, ..., α_n} \subset A[/math] such that [math]E /subset (/bigcup^n)_(i=1)G_α[/math]
A point is connected to a point if there is a path (obviously continuous) that connects them.
A connected set is a set where every point in it is connected to all other points of the set.
Inuitive and a rigorous definition in almost all contexts
A set [math]S[/math] is compact if and only if every sequence [math]s:\mathbb{N}\to S[/math] has a subsequential limit in [math] S[/math]
>>7973281
I think I like this the path-connected definition of connectedness the most
Any open cover has a finite subcover.
>>7973291
Sequential compactness (the definition you gave) isn't equivalent to compactness in general topological spaces (only in metric spaces).
>>7973311
Metric topology = best topology
>>7973281
Path connectedness isn't the same thing as connectedness, not even in metric spaces. The topologist's sine curve is the standard example of a connected but not path connected space. A topological space is connected if the only clopen subsets are the empty set and the whole space.
>>7973313
Zariski topology is best topology. Hausdorff spaces are for losers.
>>7973320
regular locales are the best
>>7973320
I never understood the fascination with primes (sorry I'm not far enough in math to really understand a prime ideal or w/e)
>>7973330
I've done that and it still isn't that interesting
is there a way to study topology without resorting to imaginary things like infinite sets?
>>7974286
I'm pretty sure there is intuitionistic topology.
>>7973330
The Nullstellensatz give a correspondence between points in [math]\mathbb C^n[/math] and maximal ideals of [math]\mathbb C[x_1,\dotsc,x_n][/math]. Interesting things happen if you extend this to prime ideals.
>>7974286
predicative locale theory ?