Search the corpus

    UNL topology workshop

    SRC-TOP-WORKSHOP

    Area
    Topology
    Status
    draft
    Source
    Contents

    153 problems.

    Day 1 Compactness

    1. A compact subset of a \(T_2\) space is closed
    2. A closed subset of a compact space is compact
    3. Cartesian products of locally compact spaces (closure formulation)
    4. A continuous bijection from compact to Hausdorff is a homeomorphism
    5. A compact subset of a product with the indiscrete two-point space
    6. A finite intersection of closed sets contained in a prescribed open neighborhood

    Day 2 Connectedness and Path Connectedness

    1. A space is connected if and only if its only clopen subsets are the empty set and itself
    2. Path-connected spaces are connected
    3. A continuous image of a path-connected space is path connected
    4. Removing a product of proper subsets from a product of connected spaces
    5. A union of connected subsets with nonempty intersection is connected

    Day 3 Separation Axioms

    1. Metric spaces are Hausdorff
    2. \(T_4\Rightarrow T_3\Rightarrow T_2\Rightarrow T_1\Rightarrow T_0\)
    3. A space is \(T_1\) if and only if the identity to the finite-complement topology is continuous
    4. A Hausdorff quotient without a countable basis
    5. A local closure condition implies regularity
    6. A separable metric space is second countable
    7. Completely normal spaces and normal subsets
    8. Countable dense products in the product topology

    Day 4 Counterexamples and Assorted Problems

    1. A neighborhood characterization of closure
    2. A continuous bijective open (or closed) map is a homeomorphism
    3. Spaces in which every open set is compact; a Hausdorff such space is finite
    4. The compact-complement topology is coarser than a Hausdorff topology
    5. A compact union of two Hausdorff subspaces
    6. Projection from a product with a compact factor is closed
    7. The unit sphere quotient by height fibers is an interval

    Day 7 Fundamental Groups

    1. A 2-complex with fundamental group \(\langle x,y\mid x^2yxy\rangle\)
    2. Nonhomeomorphic spaces with isomorphic nontrivial fundamental groups
    3. Fundamental group of the torus \(T^2=S^1\times S^1\)
    4. Fundamental groups of punctured products and a wedge of projective planes
    5. Seifert–van Kampen and the spheres
    6. A complete bipartite graph with 2-cells attached along four-edge cycles
    7. The fundamental group of three-dimensional space with its axes removed
    8. A quotient identifying points of a torus and a sphere
    9. The fundamental group of the triangular parachute
    10. A torus with a disk attached along the diagonal
    11. A torus with a disk attached along one circle factor
    12. An index-four subgroup and covering space of a free group

    Day 8 Covering Spaces

    1. The real line is the universal cover of the circle
    2. \(S^2\vee S^2\) does not cover \(S^2\)
    3. Uniqueness of lifts into a covering space
    4. The pullback of a covering map is a covering map
    5. The Klein bottle has a nonabelian fundamental group
    6. All fibers of a cover over a connected base have the same finite cardinality
    7. A pathwise continuity criterion for a lift into a cover
    8. A compact set meets each fiber of a cover in finitely many points
    9. Covering maps are open maps
    10. The universal cover and an index-two subgroup of the infinite dihedral group
    11. Connected covering spaces of the sphere with antipodal poles identified

    Day 9 Homology

    1. Homology of \(S^n\) for all \(n\ge 0\)
    2. Homology of \(T\times[-1,1]\) with each end \(T\times\{\pm 1\}\) collapsed to a point
    3. Fundamental group and homology of the mapping torus of \(z\mapsto z^4\) on \(S^1\)
    4. Fundamental group and homology of \(\mathbb{R}^4\setminus S^1\)
    5. Homology of the unit sphere union the interval from \((0,0,-1)\) to \((0,0,1)\)
    6. Homology of the complement of two unlinked circles in \(\mathbb{R}^3\)
    7. Homology of a wedge sum by Mayer–Vietoris
    8. \(\Delta\)-complex structure and homology of a 2-simplex with its vertices identified

    Legacy packet — Homotopy and retractions

    1. A retraction induces a surjection on fundamental groups
    2. A deformation retraction induces an isomorphism on fundamental groups
    3. Every map from RP^2 to the torus is null-homotopic
    4. Maps \(X\to S^1\) are nullhomotopic when \(\pi_1(X)\) is finite
    5. The boundary torus is not a retract of the solid torus
    6. Compositions of two self-maps of the circle are homotopic
    7. A homeomorphism of the closed disk preserves its boundary
    8. No map from the torus to \(S^1\vee S^1\) has a left inverse
    9. The boundary sphere is not a retract of the ball
    10. Presentation complex and universal cover for the cyclic group of order two
    11. Presentation complex and universal cover for the cyclic group of order six

    Revised packet — Compactness

    1. A compact subset of a \(T_2\) space is closed
    2. A closed subset of a compact space is compact
    3. Continuous images of compact sets are compact
    4. Nested nonempty closed subsets of a compact space have nonempty intersection
    5. A compact topology on \(\ZZ\) from cofinite sets and sets omitting \(0\)
    6. A strictly finer topology on a compact Hausdorff space is Hausdorff but not compact
    7. Compactness of the one-point compactification of \((0,1)\times(0,1)\)
    8. Normality and compactness under a coarser topology
    9. A point and a disjoint compact set in a Hausdorff space can be separated by opens
    10. Disjoint compact subsets of a Hausdorff space can be separated by opens
    11. Cartesian products of locally compact spaces (closure formulation)
    12. A continuous bijection from compact to Hausdorff is a homeomorphism

    Revised packet — Connectedness and path connectedness

    1. A space is connected if and only if its only clopen subsets are the empty set and itself
    2. Path-connected spaces are connected
    3. The union of connected sets is connected if one meets the closure of the other
    4. \(\RR\) and \(\RR^2\) are not homeomorphic
    5. Contractible spaces are path-connected
    6. Joining a connected set to one side of a separation of its complement
    7. Local connectedness via open components, and preservation under quotient maps
    8. A connected locally path-connected space is path-connected
    9. Removing a product of proper subsets from a product of connected spaces
    10. Interior of a product equals the product of interiors
    11. Identifying one point from each of two disjoint connected spaces yields a connected quotient
    12. Open connected subsets of \(\RR^n\) are path-connected

    Revised packet — Homotopy and retractions

    1. A retraction induces a surjection on fundamental groups
    2. A deformation retraction induces an isomorphism on fundamental groups
    3. Maps \(X\to S^1\) are nullhomotopic when \(\pi_1(X)\) is finite
    4. No retraction from \(S^2\) onto an equatorial circle
    5. \(q\times\mathrm{id}_I\) is continuous; the cone \(CX\) deformation-retracts to a point
    6. A map with left and right homotopy inverses is a homotopy equivalence
    7. No map from the torus to \(S^1\vee S^1\) has a left inverse
    8. The unit sphere quotient by height fibers is an interval
    9. Continuity via \(f(\overline{A})\subseteq\overline{f(A)}\), and equality when \(f(\overline{A})\) is closed
    10. Collapsing the disk fibers of \(S^1\times D^2\) yields \(S^1\)

    Revised packet — Fundamental group

    1. A 2-complex with fundamental group \(\langle x,y\mid x^2yxy\rangle\)
    2. Nonhomeomorphic spaces with isomorphic nontrivial fundamental groups
    3. Fundamental group of the torus \(T^2=S^1\times S^1\)
    4. Fundamental group of \(\RR^3\) minus the three coordinate axes
    5. Cell structure and fundamental group of a torus and sphere with two torus points identified to a sphere point
    6. A genus-2 surface is not a retract of the region it bounds
    7. Fundamental group of the triangular parachute
    8. Surjectivity of \(i_*\) on \(\pi_1\) lets paths with endpoints in \(A\) be homotoped into \(A\)
    9. A torus with a disk attached along the diagonal
    10. Fundamental group of \(\RR^2\) minus three points
    11. An index-3 subgroup of a finitely presented group is finitely presented
    12. An index-four subgroup and covering space of a free group

    Revised packet — Covering spaces

    1. The real line is the universal cover of the circle
    2. \(S^2\vee S^2\) does not cover \(S^2\)
    3. Deck transformations sending \(y_0\) to \(y_1\) iff \(p_*(\pi_1(\widetilde{X},y_0))=p_*(\pi_1(\widetilde{X},y_1))\)
    4. The fundamental group of the Klein bottle is nonabelian, via covering spaces
    5. The pullback of a covering map is a covering map
    6. A covering space restricts to a subspace, with \(\ker\iota_*\subseteq\operatorname{im}q_*\)
    7. Products of covering spaces, and the universal cover of \(S^1\times D^2\times S^1\)
    8. A covering space of a Hausdorff space is Hausdorff
    9. A finitely presented group has finitely many index-3 subgroups
    10. An index-3 subgroup of a finitely presented group is finitely presented
    11. If \(i_*:\pi_1(A)\to\pi_1(X)\) is surjective, points in a fiber of a covering are joined in \(p^{-1}(A)\)
    12. A map of path-connected covering spaces over the same base is a covering map
    13. A finitely presented group has finitely many subgroups of index \(4\)

    Revised packet — Homology

    1. Homology of \(S^n\) for all \(n\ge 0\)
    2. Singular homology of the \(2\)-torus via Mayer–Vietoris
    3. Simplicial homology of \(S^2\) with three points identified
    4. Simplicial homology of three disks with identified boundaries
    5. Euler characteristic and homology of \((\Delta^5)^{(2)}\)
    6. Nonvanishing \(H_4\) forces a 4-simplex, and \(H_4\cong\mathbb{Z}/7\mathbb{Z}\) forces a 5-simplex
    7. Homology of the hexagon quotient with labels \(ababab\)
    8. Homology of a torus attached to a cylinder along a circle
    9. Homology of \(\Delta^2\) with vertices identified
    10. \(H_i(SX)=H_{i-1}(X)\) for \(i>1\)
    11. Simplicial homology of \(\Delta^3\) after two ordered face pairings
    12. Reduced singular homology of \(S^1\times(S^1\vee S^1)\)

    Revised packet — Separation axioms, counterexamples, and definitions

    1. \(T_4\Rightarrow T_3\Rightarrow T_2\Rightarrow T_1\Rightarrow T_0\)
    2. Separating disjoint finite subsets in a Hausdorff space
    3. Normality need not pass to a coarser topology
    4. A separable metric space is second countable
    5. Completely normal spaces and normal subsets
    6. A local closure condition implies regularity
    7. Every compact metrizable space has a countable basis
    8. The compact-complement topology is coarser than a Hausdorff topology
    9. Interior of a product equals the product of interiors
    10. Path components are well-defined and need not be open or closed
    11. \(\overline{U}\setminus U\) is nowhere dense for \(U\) open
    12. Metacompactness is a homeomorphism invariant, and an included-point topology that is not metacompact
    13. Closed subsets of limit-point compact spaces are limit-point compact
    14. The product of two locally connected spaces is locally connected

    Browse these problems in source order