157 problems.
- Metric spaces are Hausdorff
- Connectedness versus path-connectedness
- Finite products and unions of compact spaces are compact
- \([0,1]\) is compact
- A sequentially compact metric space is totally bounded
- Compact subsets of Hausdorff spaces are closed
- Nowhere dense subspace
- Injective continuous maps from compact spaces to Hausdorff spaces are embeddings
- A quotient of a compact space is compact
- Local homeomorphisms of compact Hausdorff spaces are covering maps
- \(\mathbb{R}\) is not homeomorphic to \([0,\infty)\)
- Heine–Cantor theorem
- \([0,1]\) is compact without Heine-Borel
- \(\mathbb{R}/\mathbb{Q}\) has the indiscrete topology
- A metrizable space is compact if and only if it is sequentially compact
- Dense subspace
- \(\mathbb{R}\) is separable
- An injective continuous map from a compact space to a Hausdorff space is an embedding
- Compact subsets of metric spaces are bounded
- Closed subsets of compact spaces are compact
- Separable space
- \(S^n\) is not a strong deformation retract of \(B^{n+1}\)
- A continuous real-valued function on a compact space attains its bounds
- The diagonal map \(\Delta(x)=(x,x)\) is continuous
- \([0,1]\) is compact
- Intermediate value theorem
- Compact metric spaces are complete
- Compactness and limit-point compactness
- Interior, isolated, and limit points
- \(X\) is connected if and only if its only clopen subsets are \(\emptyset\) and \(X\)
- A quotient of a compact space is compact
- In a locally path-connected space, open subsets are locally path-connected, connectedness is equivalent to path-connectedness, and path components are the open connected components
- No fixed points implies homotopic to antipodal
- Compact topological space
- Continuous maps from compact spaces to Hausdorff spaces are closed
- Compact subsets of metric spaces are bounded
- Compact versus sequentially compact spaces
- Compact Hausdorff spaces are metrizable if and only if second-countable
- The continuous image of a compact space is compact
- The interval \([0,1]\) is connected
- Homology of product of spheres
- A sequentially compact space is totally bounded
- Closed subsets of compact spaces are compact
- \(X\) is Hausdorff if and only if \(\Delta(X)\) is closed in \(X\times X\)
- A closed subset of a Hausdorff space need not be compact
- A compact subset of a Hausdorff space is closed
- Every indiscrete space is separable
- A compact subset of a Hausdorff space is closed
- Compact versus sequentially compact spaces
- Continuous bijections from compact spaces to Hausdorff spaces are homeomorphisms
- Closed subsets of compact spaces are compact
- A closed subset of a compact space is compact
- Closure of a subspace
- The continuous image of a compact space is compact
- The minimal uncountable order is not separable
- Finite products and unions of compact spaces are compact
- The continuous image of a compact space is compact
- A set open in a subspace but not in the ambient space
- Definition of a basis for a topology
- Continuous maps between compact Hausdorff spaces are proper
- Closed subsets of compact spaces are compact
- Every countable discrete space is separable
- \(\RR\) with the cofinite topology is compact
- Infinite sets with the cofinite topology are compact
- The subspace of \(\RR^2\) with at least one rational coordinate is connected
- A compact Hausdorff space is metrizable iff it is second-countable
- Compact, limit-point compact, and sequentially compact coincide for second-countable Hausdorff or metric spaces
- A continuous map of metric spaces is uniformly continuous on compact subsets
- The closure is the smallest closed set containing \(A\)
- Sets between a connected subspace and its closure are connected
- \(\mathrm{cl}_X(\bigcup_i A_i)=\bigcup_i\mathrm{cl}_X(A_i)\)
- \(X\times Y\) is Hausdorff if and only if \(X\) and \(Y\) are Hausdorff
- The cofinite topology on \(\RR\) is compact
- Extreme value theorem
- Sequential compactness
- A continuous map from a compact space to a Hausdorff space is closed
- Compact subspaces of Hausdorff spaces are closed
- Six topologies on \(\mathbb{R}^{d}\)
- In a compact Hausdorff space, closed sets are exactly the compact sets
- Product, subspace, and quotient topologies
- Continuous images of connected sets are connected
- Degree of a non-surjective map \(S^n\to S^n\) is zero
- \(\pi_1(S^1\vee S^1)\)
- Connected versus locally connected spaces
- One-point compactification is compact; path-connected implies connected
- Maps \(S^1\to X\) extend over \(D^2\) when \(\pi_1(X)=0\)
- The Möbius band is homotopy equivalent to \(S^1\times I\)
- Cellular homology and cohomology of \(\RP^2\)
- The Möbius strip deformation-retracts onto its middle circle, with \(\pi_1\cong\ZZ\) generated by that circle and with boundary twice the generator
- \(\langle(1,0),(0,p)\rangle\) is an index-\(p\) subgroup of \(\ZZ\times\ZZ\)
- \(S^2\) minus two points deformation retracts onto \(S^1\)
- A manifold is orientable if \(\pi_1\) has no subgroup of index \(2\)
- Deformation retract of \(\RR^3\setminus S^1\) onto \(S^2\vee S^1\)
- \(T\cup_f D^2\) with \(f\) attaching inside the torus
- The torus does not cover \(\RP^2\)
- \(\pi_1(X\times Y)\cong\pi_1(X)\times\pi_1(Y)\)
- Homology of the Klein bottle via Mayer–Vietoris
- \(\pi_1(S^1 \times S^1)\)
- Simplicial homology of \(S^2\)
- Orientable covers of non-orientable manifolds have even or infinite degree
- Lift multiplication to the universal cover of a topological group
- \(S^n\) minus \(k\) points is a wedge of \(k-1\) spheres
- Attaching a \(2\)-cell along homotopic maps yields homotopy-equivalent spaces
- \(\pi_1\) and \(H^*\) of \(S^1\) with \(2\)-cells attached via \(z^3\) and \(z^5\); not homeomorphic to \(S^2\)
- \(\pi_1(S^1\vee S^1)\) and \(\pi_1(S^1\times S^1)\)
- Products of Hausdorff spaces; \(\RR\) versus \([0,\infty)\)
- The surface with polygonal symbol \(xyzxy^{-1}z^{-1}\)
- The infinite \(3\)-regular tree as a universal cover
- Closed versus compact subsets of Hausdorff spaces
- Cohomology of \(S^2 \vee S^2 \vee S^4\)
- \(\ZZ\)
- Cayley graph of \(\ZZ \times \ZZ\) as a covering space
- The topologist’s sine curve is connected but not path-connected
- The torus as a double cover of the Klein bottle
- For nonempty \(X,Y\), \(X\times Y\) is compact iff \(X\) and \(Y\) are compact
- Homology of \((S^1\times S^1)\vee S^1\)
- Covering maps inject on \(\pi_1\) and are homeomorphisms if they induce \(\pi_1\)-isomorphisms
- Non-surjective maps \(X\to X^n\) are nullhomotopic
- \(\pi_i(T^n)=0\) for \(i\geq 2\)
- van Kampen’s theorem, and \(\pi_1\) of glued tori, the Klein bottle, and wedges
- Compact connected surfaces with nonempty boundary are homotopy equivalent to a wedge of circles
- \(S^2\) with a disk attached along the equator
- \(\QQ\)
- Genus of an \(n\)-sheeted cover of a closed surface
- Interior of a product versus product of interiors
- \(\pi_1(S^n)\) by van Kampen on the complements of the north and south poles
- Continuous images of compact spaces, and compact metric spaces are complete
- Closed subsets of compact spaces are compact
- Generators of covering subgroups of \(S^1\vee S^1\)
- Open subsets of compact spaces
- Induced maps on \(\pi_1(S^1)\) of \(z\mapsto z^n\), antipodal, and \(\sin\) maps
- \(H^*(S^2\times S^2;\ZZ)\) via Künneth
- Here we examine
- Simplicial homology of a figure-eight graph
- Paths with common endpoints in a simply connected space are homotopic
- \(S^2\times\RP^4\) is not homotopy equivalent to \(S^4\times\RP^2\)
- Connected covering spaces of \(\RP^2\vee\RP^2\)
- Covering spaces and deck transformations of the Klein bottle, Möbius band, and \(\RP^2\times S^1\)
- Regular covering spaces and a non-regular cover of \(\Theta\)
- Every compact metrizable space has a countable basis
- Contractible if and only if the identity is nullhomotopic
- \(\chi(\Sigma_g+\Sigma_h)=\chi(\Sigma_g)+\chi(\Sigma_h)-2\)
- Degree of the antipodal map, a CW structure on \(\RP^n\), and even-dimensional coverings
- Winding number of \(8z^4+4z^3+2z^2+z^{-1}\) on \(S^1\)
- Even-degree orientable covers of non-orientable manifolds, and \(\pi_1\) without index-\(2\) subgroups
- The antipodal map on \(S^n\) is homotopic to the identity when \(n\) is odd
- Strict implications \(T_4\Rightarrow T_3\Rightarrow T_2\Rightarrow T_1\Rightarrow T_0\)
- A space is connected iff its only clopen subsets are \(\emptyset\) and \(X\)
- Homology of \(\RP^2\) via Mayer-Vietoris
- A non-surjective map into \(S^n\) is nullhomotopic
- Homotopy \(H(x,t)=P(tf(x)+(1-t)g(x))\) when \(f(x)\neq -g(x)\)
- Homology of two \(2\)-spheres glued along their equators
- Homotopy types of the letters of the alphabet
- \(\pi_1(\bigvee S^1)\) is free, and \(S^2\not\cong S^3\)
- When compact sets are closed, and \(\overline{A\times B}=\overline{A}\times\overline{B}\)
- Fundamental group of \(S^2\) with equatorial antipodes identified
- Closed sets covering a connected space with connected intersection are connected