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In any topological space X, the empty set is open by definition, as is X. Since the complement of an open set is closed and the empty set and X are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact. The closure of the empty
Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology. Any space carrying the cofinite topology is compact.
Note that one formulation of AC is that the Cartesian product of a family of nonempty sets is nonempty; but since the empty set is most certainly compact, the proof cannot proceed along such straightforward lines. Thus Tychonoff's theorem joins several other basic theorems (e.g. that every vector space has a basis) in being equivalent to AC.
Empty set; Finite set, Infinite set; Countable set ... Closed set; Open set; Clopen set; F σ set; G δ set; Compact set; Relatively compact set; Regular open set ...
Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T 1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable ...
In particular, every sequence has a convergent subsequence (the whole sequence or any other subsequence), thus X is sequentially compact. The interior of every set except X is empty. The closure of every non-empty subset of X is X. Put another way: every non-empty subset of X is dense, a property that characterizes trivial topological spaces.
Compact elements cannot be approximated by elements strictly below them. On the other hand, it may happen that all non-compact elements can be obtained as directed suprema of compact elements. This is a desirable situation, since the set of compact elements is often smaller than the original poset—the examples above illustrate this.
σ-compact. A space is σ-compact if it is the union of countably many compact subspaces. ... a space is connected if the only clopen sets are the empty set and itself.