Metric and Topological Spaces

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Exercises 8

  1. Prove that every subset of a Hausdorff space is Hausdorff in the subspace topology.
    1. If X and Y are Hausdorff, prove that X cross Y is Hausdorff.
    2. Prove or disprove: The image of a Hausdorff space under a continuous map is Hausdorff.
    3. If ~ is an equivalence relation on a Hausdorff space X, is the space X/~ with the identification topology always Hausdorff ?

    Solution to question 1

  2. Show that the topology on R whose basis is the set of half-open intervals [a, b) is normal.

    Solution to question 2

  3. A T1 space is one in which for every pair of points x noteq y there is an open set containing x but not y.
    Prove that a space is T1 if and only if every singleton set {x} is closed.
    Prove that the only T1 topology on a finite set is the discrete topology.

    Solution to question 3

  4. Prove or disprove:
    1. The product of connected spaces is connected.
    2. If X is connected, then X/~ is connected (where ~ is an equivalence relation).

    Solution to question 4

  5. Let Y be the space {a, b} with the discrete topology. Prove that a space X is connected if and only if the only continuous maps from X to Y are the two constant maps which map the whole of X to either a or b.

    Solution to question 5

  6. If A is connected, prove that the closure cl(A) is also connected. Deduce that the components of a space are always closed subsets.
    Is the interior int(A) always connected ?

    Solution to question 6

  7. Is the space R with the topology of Question 2 a connected space? What are its components?
    What are the components of R with:
    1. the co-finite topology,
    2. the co-countable topology.

    Solution to question 7

  8. If a, b are points in a topological space, define a tilda b if there is a connected subset of X containing a and b. Prove that tilda is an equivalence relation.
    If a, b are points in a topological space, define a tilda2 b if there is a path in X connecting a and b. Prove that tilda2 is an equivalence relation.
    Deduce that pathwise connectedness implies connectedness.

    Solution to question 8

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JOC February 2004