Metric and Topological Spaces

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Exercises 6

  1. If U subset A subset X, prove that U is closed in the subspace topology on A if and only if U = A intersect Z for Z a closed subset of X.
    Prove or disprove the following:
    1. The interior of U in the subspace topology on A is equal to the interior of U in the topology on X,
    2. The closure of U in the subspace topology on A is equal to the closure of U in the topology on X.

    Solution to question 1

  2. Let curlyT1 and curlyT2 be the subsets of the natural numbers N defined by:
    U belongs curlyT1 if either U = empty or N - U is finite,
    U belongs curlyT2 if either 1 notbelongs U or N - U is finite.
    Prove that curlyT1 and curlyT2 are topologies on N.
    Let f be the identity map on N and let g be the map from N to N defined by
    g(n) = 1 if n is odd; g(n) = 1 + n/2 if n is even.
    Determine whether f and g are continuous either as maps from (N, curlyT1) to (N, curlyT2) or as maps from (N, curlyT2) to (N, curlyT1).

    Solution to question 2

  3. Let A subset Y and B subset Y so that A cross B subset X cross Y. Prove that:
    1. cl(A) cross cl(B) = cl(A cross B)
    2. int(A) cross int(B) = int(A cross B)
    where cl denotes the closure and int denotes the interior.

    Solution to question 3

  4. A set curlyS of subsets of a topological space X is called a sub-basis for the topology if every open set can be written as a arbitrary union of finite intersections of sets in curlyS.
    Show that a function f from a topological space X to a topological space Y is continuous if and only if f -1(U) is open for every set U in a sub-basis for the topology on Y.
    Prove that the set of all unbounded open intervals of R forms a sub-basis for the usual topology on R which is not a basis.
    Prove that a sub-basis of the product topology on X cross Y is the set of subsets of the form U cross Y and X cross V for U belongs curlyTX and V belongs curlyTY .

    Solution to question 4

  5. Prove that the set of all epsilon-neighbourhoods of rational points of R with epsilon also rational, forms a basis for the usual topology on R. Deduce that the usual topology on R has a countable basis. Prove that the discrete topology on R does not have a countable basis.

    Solution to question 5

  6. Let X be the open unit square and let Y be the open unit quadrant, each with their topology as subsets of R2. Prove that the map which takes the point (x, y) (in Cartesian co-ordinates) to (x, 1/2py) (in polar co-ordinates) is a homeomorphism.
    Prove that this homeomorphism cannot be extended to a homeomorphism between the closed unit square and closed unit quadrant.
    Show, however, that the closed unit square and closed unit quadrant are homeomorphic.

    Solution to question 6

  7. Prove that [0, 1) cross (0, 1) and [0, 1) cross [0, 1] are homeomorphic subspaces of R2.

    Solution to question 7


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JOC February 2004