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 , exists , forall , forall ,
Meaning: The elements of any net have a common limit point 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)
By using the equivalent limit <==> image net finer, compact can also be represented by replacing any net of with any source_net_space -> target_topology_space function. Although this introduces an "extra" domain source_net_space
According to the definitions of limit points and closed sets, compact is equivalent to: forall net of ,
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
==>