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.
The key points in the solutions of the two given problems in topology are provided, and complete, detailed solutions are given in an attached .doc file.
1. Given an infinite set X, the cofinite topology T on X, and the filter F generated by the filter base consisting of all cofinite subsets of X, the set of points of X to which F converges was found.
2. A complete, detailed proof is given of the fact that a space X is compact if and only if every open cover has an irreducible subcover; a counterexample (i.e., an example of a non-compact space X, and an open cover of X that has no irreducible subcover) is also given, together with a justification for the counterexample.
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- Attached file(s):
- Filter&IrreducibleCover-Solutions.doc
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