A Little Topology

Convex sets are closed (in a certain sense) under linear operations.
Closed sets are closed under limit operations.
Compact sets are closed and finite (in a certain sense) under limit operations.

convex set ~ concave function ~ linear operation
closed set ~ continuous function ~ limit operation

Closed sets contain all the limits of its convergent sequences. With compact sets the limits not only are contained within the set but also are insured of existence for at least one subsequence of every sequence, i.e. sequences are bounded within compact sets, while closed within closed sets.Building on the spacial property of distance in metric spaces and extending beyond to general non-metric spaces.

...open sets satisfy the axioms of topological space... 

Concept of limit. Sensible only in the context where a 'topology' e.g.distance is defined.

...a sequence is not a set of points...rather, it is a function... 

A limit is something that can be approached by a sequence.

...limit points are limits of sequences...

...a closed set contains all its limit points...

...a closed set is a set which is closed under the limit operation... 

Continuity of a function is the continuity of limit operation before and after the function.
Metric functions are continuous by definition because of triangular inequality.
Closed and bounded = compact