general-topology

  1. Let $\mathbb{D}^3=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2\leq 1\}$. Define $X=\mathbb{D}^3/\sim$ with the relation $(x,y,z)\sim(-x,-y,-z)$

  2. Let $A$ be a finely generated Abelian group. Prove that there is a topological space $X$ with a basic point $x_0$ such that $\pi_1(X,x_0)\cong A$.

  3. Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

  4. Unions of topological dynamical systems - a counter example
  5. Does every torus $T\subset S^3$ bounds a solid tours $S^1\times D^2\subset S^3$?
  6. Does set with Lebesgue-Mass nonzero have almost surely an open subset

  7. Why are continuous functions the "right" morphisms between topological spaces?
  8. Is below subspace is connected?

  9. Is the converse of Tychonoff's theorem true?

  10. Is the following embedding of $k5$ on torus not a 2-cell embedding? If not, how can we get the 2-cell embedding of $k5$ on the torus?
  11. Definition of disconnected?

  12. About spaces $\Bbb R^n/\!\sim$ in which certain coordinate permutations do not matter.
  13. Prove isomorphism in infinite-dim condition

  14. Action of universal covering deck transformations group
  15. some confusion in cofinite topology..
  16. Convergence in a Topological Space

  17. Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous, show that $\{x:f(x)\leq g(x)\}$ is closed in $X$

  18. Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$.

  19. Definition of topology induced by seminorms

  20. Can we recover a topology from knowing about continuity of maps from compact spaces?

  21. $X$ compact convex subset of $\mathbb{R^n}$. $f:X\to X,f(Fr(X))\subset Fr(X)$. $f|_{Fr(X)}$ is homotopic with identity on $Fr(X)$. Show that $f(X)=X$
  22. The Weak topology on an infinite-dimensional space is not metrizable
  23. Coequalizers are quotient maps

  24. The subset $[a, b)$ of $\Bbb R$ is neither open nor closed.
  25. Help Interpreting a Problem Concerning the Union of Connected Sets
  26. Two tori $\mathbb C/L$ and $\mathbb C/L'$ are isomorph if $L=L'$

  27. lower limit topology to the metric topology

  28. Is this space $T_1$ and satisfies property $P$?.
  29. Proving that $\mathbb{Q}$ is neither open or closed in $\mathbb{R}$
  30. Hausdorff space if no net converges to two different values

  31. Why is empty set an open set?
  32. Filters and surjectivity of functions
  33. Nerve Theorem: Is the finite union of closed convex sets triangulable?

  34. Topologies of Matrix Groups & Algebras
  35. Why is unwinding a string not a continuous transformation?
  36. Subfunctor of a hom functor in a Boolean topos
  37. Prove that Voronoi cells are path-connected

  38. Completion with respect to stronger norm is no subset?

  39. Example of infinite , compact, path connected metric space $X$ which is homeomorphic to $X \times X$?
  40. Closure as a map

  41. How to use a continuous function on $[0,1]$ to prove another set is connected?

  42. Continuity of a function from a one-point compactification to a cone

  43. Prove two metrics are equivalent?

  44. Product Topology/Continuity Confusion
  45. Quotient space and quotient map

  46. topology generated

  47. Is every open set a union of (any number of) sets that are homeomorphic to the entire space?

  48. To get a amall strip near the boundary
  49. Decomposing a topological space into closed subspaces

  50. (Topology) In the real line $R^1$ a set is open if and only if it is a union of open segments (a,b).
  51. Imagining inside of complement set
  52. Find the boundary and interior of $C= A \cup B$?

  53. Soft question - a subset of a Hilbert space endowed with subspace topology
  54. "Inherits as a Subspace" in Toplogy

  55. Uniqueness in Baire property representation for compact Hausdorff spaces

  56. Is below set is open In $\mathbb{R^2}$

  57. Induced Borel $\sigma$-algebra.

  58. Show that a function is continuous in one variable.
  59. Prob. 9, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: If $X$ and $Y$ are connected and if $A$ and $B$ are proper subsets of $X$ and $Y$, resp., then

  60. Find $f : [0,1] \to \mathbb R$ s.t. $\{x \in [0,1]$, for all neighborhood of $x$, there exist $y$ and $z$ in it s.t. $f(y)f(z)<0 \}\ $ is uncountable

  61. Find a topological space $X$ which is connected but has three path components.

  62. A question regarding complements of dense subsets.

  63. Let $\mathbb{R}$ with cofinite topology then infinite subset of $\mathbb{R}$ is?

  64. Show that $(\Bbb R^2,T_1)$ and $(\Bbb R^2,T_2)$ are not homeomorphic

  65. Why Does the Definition of a Topology via Neighborhoods Include This Axiom

  66. Is $X$ a $T_1$ space? (i.e., given a pair of distinct points in $X$, does each one of them have a neighborhood not containing the other?)
  67. Can someone help me understand this proof of a modified version of Brouwer's theorem?

  68. Unit ball in $l^{2}$ is bounded and closed but not compact.

  69. Brouwer theorem (specific version)

  70. Definition of 'saturated' set?

  71. Associated Bundles: Necessity of Action on Left and Right?

  72. Possible groups for fundamental group of $\mathbb{R}$
  73. Homeomorphism of $K$ and $K\cup \{0\}$
  74. every open set in the extended real line ($\overline{\mathbb R}$) is a countable union of segments

  75. Prob. 8, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: Is $\mathbb{R}^\omega$ connected in the uniform topology?

  76. Prob. 2, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: If $J$ is a well-ordered set and if $\left\{ X_\alpha \right\}_{\alpha \in J}$ . . .

  77. Difference between topology and sigma-algebra axioms.
  78. To introduce a metric on $\mathbb{R}$, such that $f(x) = x^2$ is uniformly continuous

  79. What is the relationship between topological gluing and quotient space?
  80. Show that $\mathcal{A}$ is dense in $C([0,2016])$.
  81. Example of a non-compact manifold

  82. Simple question on continuos function & Hausdorff space
  83. Which of the following sets are necessarily perfect, connected?

  84. Is the complement of a bounded set bounded?
  85. Necessary/sufficient condition for directional derivative to be zero (infinite-dimensional optimization)
  86. Given a metrizable topological space $(X,\tau)$, how to construct a metric that induces $\tau$?
  87. Proof verification: topology on the extended real line

  88. from weakly sequentielly continuous non linear map to continuous map

  89. Characterisation of Borel hierarchy.

  90. some confusion in symbolic general topology
  91. Homeomorphic function spaces arising from homeomorphisms between domains and images?
  92. Connected set in R^2
  93. Strongly separating hyperplane
  94. Is $\overline{A}$ path-connected?
  95. Why a metric tensor is considered a tensor as opposed to a matrix?

  96. Given an element $ \ (a,b) \in A \ $ describe its equivalence class $ \ [(a,b)] \ $ geometrically

  97. Difference between basis and subbasis of a topology
  98. Show that the addition function for two elements in $\mathbb{R}^\omega$ is continuous

  99. topological properties of sets $\textit{Iso}(X,d)$ , $\textit{Iso}_{\epsilon} (X,d)$

  100. Do 2 sets exist that are neither open nor closed, but their union is both open and closed.