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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.
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.
Every CG-3 space is a T 1 space (because given a singleton {}, its intersection with every compact Hausdorff subspace is the empty set or a single point, which is closed in ; hence the singleton is closed in ).
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 ...
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.
Closure of compact not compact The set {p} is compact. However its closure (the closure of a compact set) is the entire space X, and if X is infinite this is not compact. For similar reasons if X is uncountable then we have an example where the closure of a compact set is not a Lindelöf space. Pseudocompact but not weakly countably compact