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Co-Finite Topology

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2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?

3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.

4. Let X be a set and F is separating collection of functions f: X --> Y_f, each from X into a topological space Y_f. Prove that X with the weak topology by F is Hausdorff.

5. Explain how we can think of the unit sphere (a subspace of IR^3 with its usual topology) as a subset of the Hilbert cube.

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Solution Preview

Problem 2.
We'll prove that no two open setsare disjoint, which implies the fact that X is not Hausdorff (see attached).

Problem 3.
First assume X is uncountable.
Let there exist an open set A, in X. Since A is open, XA is countable and closed. Since X is uncountable, A must also be uncountable, as XA is countable. ...

Solution Summary

In a co-finite topology is a subset Hausdorff? Are compact sets closed? Explain how could we think of the unit sphere as a subset of a Hilbert cube.

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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.

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