general-topology

  1. The unit ball in $L^{\infty}$ is not weakly sequentially compact.

  2. Homeomorphism to Cantor set and surjection onto $[0,1]$
  3. Showing that replacing functions with values preserves concavity

  4. Certain Subset of Sorgenfrey Plane is Closed
  5. Find all $\alpha$ for which $T_\alpha$ is well-defined, injective, $J o T_\alpha $ is a contraction, $T_\alpha$ is a homeomorphism.

  6. Computing Cohomology of Product

  7. Your favourite application of the Baire Category Theorem
  8. If $X$ is Hausdorff and the quotient map $q\colon X\to X/\mathord{\sim}$ is closed, must $\sim$ be closed in $X\times X$?
  9. Intersection of collection of open sets
  10. For all polyomial $P$ in $\mathbb C[X]$ there exists a norm such that $(X^n)$ tends to $P$

  11. Prob. 1 (a), Sec. 24, in Munkres' TOPOLOGY, 2nd ed: Is removal of a single point each from two homeomorphic spaces still leaves them homeomorphic?

  12. Why does this happen in the definition of compactness?

  13. Integer Spheres and a Real Sphere
  14. The given set is closed in $\mathbb{R}$?

  15. Do homeomorphisms on $\mathbb{R}\cup\{\pm\infty\}$ or $\mathbb{Z}\cup\{\pm\infty\}$ have zero topological entropy?

  16. Disjoint union on bundles
  17. Question on Furstenberg's proof

  18. Proof completeness of $(X,d)$ given $d$ is French railway metric

  19. A question about filters
  20. Is this proof for Theorem 16.4 Munkres Topology correct?

  21. $U,V$ are disjoint open sets in $X$ and $P\subset \overline{U}\cap \overline{V}$, then $P\cap(U\cup V)=\varnothing$
  22. General topology, prove or disprove.

  23. Prob. 4, Sec. 24, in Munkres' TOPOLOGY, 2nd ed: If an ordered set in the order topology is connected, then it is a linear continuum
  24. If $M$ is a maximal ideal in $C^{\ast}X)$, and $Z[M]$ is a $z$-filter ,is $Z[M]$ a $z$-ultrafilter?
  25. Is this set countable?

  26. Open set in order topology
  27. Order Topology on $\mathbb{Z_+}$
  28. Usual and order topology coincides

  29. Topology-Continuous

  30. On Urysohn's lemma

  31. Is $A(x) = y^{T}*x \leq b$ a closed mapping?

  32. Is there a known topological space homeomorphic to $(M\times M)/\Bbb Z_2$?

  33. Is a graph of the function connected?
  34. $g\circ f$ is open, $f$ continuous and surjective $\Rightarrow g$ is open

  35. The intersection of closure of span of infinite, linearly independent, closed, bounded, connected and disjoint subsets of $\ell^2$
  36. Let $A:l_1 \to c_0 : (x_k)_{k=1}^\infty \to (x_k^2)_{k=1}^\infty$. Is the operator A injective, surjective, continous, homeomorphism?

  37. What should be the intuition when working with compactness?

  38. Condensation points
  39. Does the family obtained by removing nowhere dense sets from open sets form a topology?

  40. Prob. 12, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: If $Y\subset X$, $X, Y$ are connected, and $A, B$ form a separation of $X-Y$,
  41. Uncountable Set with Cofinite Topology
  42. $a \in A$ and $W \in V_A(a)$ iff there is a $V \in V(a)$ such that $W=V \cap A$

  43. $C(\mathbb{Q})$ and $C^{*}(\mathbb{Q})$
  44. Take a regular coordinate ball and you get a manifold with boundary.
  45. Good reference for metric topology

  46. An example of set which is not Souslin set

  47. If $I$ and $J$ are $z$-ideals, then $IJ = I \cap J$
  48. prime and maximal ideal in $C(X)$
  49. Give a cell decomposition of $[0, 1] \subseteq \mathbb{R}$

  50. Uniform boundedness principle for lower semicontinuous functions

  51. Meager set of number in base $2$

  52. Show that set $\{x \in C^{1}[a,b] :|x(a)| \leq B_0, |x'(t)| \leq B_1\} $ is compact or not compact

  53. Total order and its order topology

  54. Order topology on a poset

  55. Order topology and subspace topology in $\mathbb R$

  56. What are the interior and acculumation points of given subsets of R
  57. Non-dense sets and their boundaries
  58. $X \setminus point$ not path-connected implies $X$ simply connected
  59. A topology finer than the final topology?

  60. Writing a Set in Terms of a Sequence of Sets

  61. Showing two topological spaces are not homeomorphic

  62. Sequence space endowed with product topology
  63. Transition Functions for Charts on Tangent Bundle

  64. Classify a surface
  65. Is the subset $\{(x,x\sin\Big(\dfrac 1x \Big)) : x >0\} \cup \{(x,y) \in \mathbb R^2 : (x+1)^2+y^2 \le 1\}$, of the plane, path connected?

  66. Functions on a cover with the property $f(U \cap V) = f(U)\,f(V)$

  67. Proving restriction of function is continuous

  68. Are open affine sets a base for the topology of a scheme?

  69. Every singleton set is open.

  70. Show that $X$ is a $T^1$ space.

  71. Connectedness and normed spaces
  72. Covering map is proper $\iff$ it is finite-sheeted

  73. A counterexample using the trivial topology.
  74. The set that we add to compactify is itself compact.
  75. Existence of continuous function $f$ on $\Bbb R$ which vanishes exactly on $A\subset \Bbb R$

  76. Are there topological manifolds with boundary that are not compact?

  77. Openness of a map of $G$-spaces
  78. Degree of covering map
  79. If a function $f: [0,1] \to \mathbb R$ is continuous, does it mean that it's uniformly continuous?
  80. The ideal generated by the identity function $i$ in $C(\mathbb{R})$

  81. Connected spaces theorem

  82. Show that the preimage of the intersection of the images of two open sets, is open.

  83. suppf as a property of a partition of unity
  84. Inclusion properties of interiors of subsets in a topological space.
  85. Topological degree of an injective mapping is $\pm 1$

  86. Suppose $M$ is a manifold of dim $n \geq 1$ and $B \subseteq M$ is a regular co-ordinate ball. Show $M \setminus B$ is a $n$-manifold with boundary

  87. Determine interior and boundary of $A\times B$
  88. A contractible space is path connected.

  89. Topology and continuity
  90. Linear mapping being not open and not closed

  91. Does there exist a continuous surjection from $\Bbb R^3-S^2$ to $\Bbb R^2-\{(0,0)\}$?
  92. An open connected subset of a Peano space is arcwise connected.
  93. Sphere with three Möbius strips glued and sphere with a handle and a Möbius strip glued
  94. Cantors Intersection theorem and convergence of cauchy sequences to more than one point in a metric superspace?

  95. Example of a weak mixing that is not Topological Mixing
  96. Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric

  97. Union of a family of compact sets
  98. When a non-constant, non-negative continuous function on Möbius strip has zero apart from its edge.

  99. Topological space lacking a point

  100. Implication about homeomorphism