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    UNL Math 871 Fall 2013 problem sets

    SRC-UNL-MATH871-FALL-2013-PROBLEM-SETS

    Area
    Topology
    Status
    draft
    Source
    Contents

    70 problems.

    PS1

    1. Subset membership under union and intersection — Munkres 1.2, parts c, g, k, m, p, q
    2. Preimage and image under injective and surjective maps
    3. Preimage preserves all set operations; image preserves unions only — Munkres 2.2; part b selected for grading
    4. Subsets of finite sets are finite
    5. Countability of \(\mathbb{Q}\)
    6. Countability of function spaces and subsets of \(\mathbb{Z}_+\) — Munkres 7.5, parts a, e, i
    7. The countable complement topology and an infinite complement collection
    8. Intersections and unions of families of topologies — Munkres 13.4; parts a, c selected for grading

    PS2

    1. Included-point and finite-complement topologies on \(\mathbb R\) are not homeomorphic
    2. The identity map between two topologies on a set
    3. Intervals homeomorphic with the unit interval
    4. Topology generated by subsets with infinite complement
    5. Countable rational-endpoint bases

    PS3

    1. Transitivity of the subspace topology
    2. Open sets in a closed interval versus open sets in the line
    3. Projections from a product are open maps — PS3.1(a), Munkres 16.4
    4. Coordinate projections from arbitrary products are open maps — PS3.1(b)
    5. Products of subspaces are subspaces
    6. The coarsest topology making a family of maps continuous
    7. Coordinate slices are imbeddings
    8. Product and cofinite topologies on \(\mathbb Z^2\)

    PS4

    1. Continuity into the included-point topology
    2. Products of closed sets are closed
    3. Restrictions of closed maps and non-closed product projections
    4. The set where f is at most g is closed — Munkres 18.8 with Y = R in the Euclidean topology
    5. Closures of intersections and differences — Munkres 17.8, parts a, b
    6. Interior and boundary of a subset — Munkres 17.19, parts a, b
    7. Boundaries and interiors of plane regions — Munkres 17.20, parts c, f
    8. Continuous images of limit points

    PS5

    1. Quotient a set by the fibers of a surjection
    2. Quotient topology on a three-point image of the line
    3. Continuous maps with continuous sections are quotient maps
    4. Inverses and compositions of homeomorphisms are homeomorphisms
    5. Composition, products, and corestrictions of open maps
    6. Four quotient-space identifications
    7. The standard genus-two surface from an octagon

    PS6

    1. Products of Hausdorff spaces are Hausdorff
    2. Subspaces of Hausdorff spaces are Hausdorff
    3. Products of Hausdorff spaces in the box and product topologies

    PS7

    1. The metric is continuous exactly for its own topology — Munkres 20.3, part a
    2. The subspace metric
    3. Separate a point from a closed set in a metrizable space
    4. Connectedness and path components of the flea-and-comb space
    5. Components and continuous maps out of the lower limit line — PS7.3(a), components and path components of the lower-limit line
    6. Components of excluded-point and included-point real lines — PS7.3(b,c)
    7. Connected fibers over a connected base connect the total space
    8. Fixed points of continuous self-maps of the interval
    9. Behavior of path connectedness under products, closures, images, and unions — Munkres 24.8; part a selected for grading

    PS8

    1. Finite unions of compact subspaces are compact
    2. Compactness and separation axioms for four real-line topologies
    3. Three quotient-space constructions from squares and disks
    4. Separating disjoint compact sets in a Hausdorff space
    5. Generalized tube lemma for compact rectangles
    6. Compact metrizable spaces are second countable
    7. Countability axioms pass through open maps — Munkres 30.12, second-countability part only
    8. Normal spaces have disjoint closure neighborhoods of closed sets
    9. The equalizer of two maps into a Hausdorff space is closed
    10. Closed subspaces of normal spaces are normal
    11. Separation properties project from products to factors — Munkres 32.2; T4 part selected for grading

    PS10

    1. Deformation retract punctured Euclidean space onto the sphere — Hatcher p.18
    2. Identify a mapping cylinder and a mapping torus
    3. A mapping cylinder deformation retracts onto its target
    4. Homotopy equivalence and homotopy are equivalence relations — Hatcher p.18
    5. Contract the equatorial loop in \(S^2\)
    6. Retracts of contractible spaces — Hatcher p.18
    7. Contractible spaces are path-connected — PS10.4
    8. Fundamental group of the two-point indiscrete space

    PS11

    1. Cancellation property for composition of paths — Hatcher p.38
    2. \(\pi_1(X)\) abelian iff basepoint-change homomorphisms depend only on endpoints — Hatcher p.38
    3. Functoriality and change of basepoint for the fundamental group

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