the original inspiration of compactness: The intersection of closed interval nets of is non-empty closed-interval-net-theorem
is a limit point of of . The net seems to converge to
But
Compare the multiplicative inverse's
does not have corresponding to the possible limit
Compare
let topological space. let
[compact] compact := forall net of ,
Meaning: The elements of any net have a common limit point set under the topology . Or, after closure, the net converges to a non-empty set or the intersection is non-empty, instead of converging to the empty set (e.g., Euclidean converges to the empty set or converges to infinity, but there are many other complex situations)
Any net can replenish all finite intersections and maintain the same limit, so for compact, the equivalent description is
compact <==>
logically equivalent to
logically equivalent to [compact-finite-open-cover]
[compact-subset] := topology-subspace compact
recall closed-in-subspace, , denoted as
compact-subset logically equivalent to
logically equivalent to
logically equivalent to [compact-subset-finite-open-cover]
compact-subset is closed under finite unions. this is easy to proof
[closed-set-in-compact-space-is-compact] compact and closed ==> compact
Proof
closed in ==> . by closed-in-subspace
Reusing compact to get and thus get compact
Hausdorff space :=
Hausdorff + compact ==> closed. At this time, compact is closed for any intersection
[continous-preserve-compact] let . is compact-subset of
Proof
Using topology-subspace, just need to handle the case
let be net of . to prove
is net of
compact ==>
The inverse image of a continuous function preserves closed . Use the property of inverse images on
surjective ==>
so , so compact
Contrapositive: Under a continuous function, the inverse image of non-compact is non-compact
[quotient-topology-preserve-compact] For quotient-topology , source space compact ==> quotient space compact. because the quotient map is continuous, it preserves compact
[product-topology-preserve-compact] product-topology preserves compact
Proof
Take a net of , need to prove or
is net of
According to compact
==>
According to the definition of closure
==>
Defined by the point-net system of product topology and the definition of closure
==>