Find the set of points of convergence of a given filter on an infinite set X with the cofinite topology. Prove that a space is compact if and only if every open cover has an irreducible subcover.

1. Let X be an infinite set, let T be the cofinite topology on X, and let F be the filter generated by the filter base consisting of all the cofinite subsets of X. To which points of X does F converge? 2. Let X be a space. A cover of X is called irreducible if it has no proper subcover. (a) Prove that X is compact if and only if every open cover of X has an irreducible subcover. (b) Give an example of a non-compact space X and an open cover of X that has no irreducible subcover.

Consider an arbitrary mapping f : X → Y. Prove the main property of the first set mapping: f(X) is a subset of Y.

Topology Sets and Functions (XXIII) Functions Consider an arbitrary mapping f : X → Y. Prove the main property of the first set mapping: f(X) is a subset of Y. See the attached file.

Consider an arbitrary mapping f : X → Y. Prove the main property of the first set mapping: A1 is a subset of A2 implies that f(A1) is a subset of f(A2).

Topology Sets and Functions (XXIV) Functions Consider an arbitrary mapping f : X → Y. Prove the main property of the first set mapping: A1 is a subset of A2 implies that f(A1) is a subset of f(A2). ... click for more

Consider an arbitrary mapping f : X → Y. Prove the main property of the first set mapping: f(∩i Ai) = ∩i f(Ai)

Topology Sets and Functions (XXVI) Functions Consider an arbitrary mapping f : X → Y. Prove the main property of the first set mapping: f(∩i Ai) = ∩i f(Ai) See the attached file.

Consider an arbitrary mapping f : X → Y. Suppose that f is a one-to-one onto. Prove the main property of the second set mapping: f – 1(φ) = φ

Topology Sets and Functions (XXVII) Functions Consider an arbitrary mapping f : X → Y. Suppose that f is a one-to-one onto. Prove the main property of the second set mapping: f – 1(φ) = φ See the attached file.