Properties related to pseudocompactness

• In order that a Tychonoff space
  Tychonoff space
  In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
  
   X is pseudocompact it is necessary and sufficient that every locally finite collection
  Locally finite collection
  In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension....
  
   of non-empty
  Empty set
  In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
  
   open set
  Open set
  The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
  
  s of X is finite. A series of equivalent conditions were given by Kerstan and Yan-Min and other authors (see the references).

• Every countably compact space is pseudocompact. For normal Hausdorff spaces the converse is true.

• As a consequence of the above result, every sequentially compact space is pseudocompact. The converse is true for metric spaces. As sequential compactness is an equivalent condition to compactness
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   for metric spaces this implies that compactness is an equivalent condition to pseudocompactness for metric spaces also.

• The weaker result that every compact space is pseudocompact is easily proved: the image of a compact space under any continuous function is compact, and the Heine–Borel theorem
  Heine–Borel theorem
  In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:*S is closed and bounded...
  
   tells us that the compact subsets of R are precisely the closed and bounded subsets.

• If Y is the continuous image of pseudocompact X, then Y is pseudocompact. Note that for continuous functions g : X → Y and h : Y → R, the composition
  Function composition
  In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
  
   of g and h, called f, is a continuous function from X to the real numbers. Therefore, f is bounded, and Y is pseudocompact.

• Let X be an infinite set given the particular point topology
  Particular point topology
  In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
  
  . Then X is neither compact, sequentially compact, countably compact, paracompact nor metacompact. However, since X is hyperconnected, it is pseudocompact. This shows that pseudocompactness doesn't imply any other (known) form of compactness.

• In order that a Hausdorff space
  Hausdorff space
  In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
  
   X is compact it is necessary and sufficient that X is pseudocompact and realcompact (see Engelking, p. 153).

• In order that a Tychonoff space
  Tychonoff space
  In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
  
   X is compact it is necessary and sufficient that X is pseudocompact and metacompact (see Watson).

See also

• Compact space
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
  
• Paracompact space
  Paracompact space
  In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....
  
  
• Normal space
  Normal space
  In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
  
  
• Realcompact space
  Realcompact space
  In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and every point of its Stone-Cech compactification is real...
  
  
• Metacompact space
  Metacompact space
  In mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement...
  
  
• Tychonoff space
  Tychonoff space
  In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....