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      Questions tagged [gn.general-topology]

      Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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      On the universal property for interval objects

      In his lecture, The Categorical Origins of Lebesgue Measure, Professor Tom Leinster mentions the following theorem: Theorem 1: (Freyd; Leinster) The topological space $[0, 1]$ comes equipped with ...
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      131 views

      Does using continued fractions work to give a homeomorphism $\mathbb{Q}^+ \rightarrow (\mathbb{Q}^+)^2$?

      Let $\mathbb{Q}$ be the set of rational numbers and let $\mathbb{Q}^+$ be the set of positive ($x>0$) rationals. I'm looking for a simple construction of a homeomorphism $\phi: \mathbb{Q} \...
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      If all length metrics are strong equivalent on a closed connected topology manifold?

      Let $M$ be a connected closed topology manifold and $d$ is a length metric (or an inner metric) on it , i.e. the distance between every pair of points is equal to the infimum of the length of ...
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      53 views

      Structure of extensions arising in Lie approximation of connected groups

      My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known: Let $G$ be a connected, locally compact, Hausdorff group, ...
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      Cardinal Invariants and Physics

      There are many applications of topology to physics, but I wonder if there is a known application of cardinal invariants to physics.
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      1answer
      141 views

      Must a locally compact, second countable, Hausdorff space support a Radon measure?

      Let $X$ be a locally compact, second countable and Hausdorff space, must there be a Radon measure on $X$ whose support is $X$? The motivation for this question comes from Anton Deitmar's paper On ...
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      260 views

      In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point?

      I previously asked In which topological spaces does the existence of a loop not contractable to a point imply there is a non-contractable simple loop also? Given the broad scope of this question I ...
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      1answer
      159 views

      In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?

      In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also? Note that ...
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      1answer
      136 views

      Is every closed subset of finite measure contained in an open subset of finite measure?

      Could someone will verify my statement: For every locally finite Borel measure on metric space and closed set $F$ with finite measure, there exists open set $U$ such that $F \subset U$ and $U$ has ...
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      In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

      I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces. If it is true that: In a Topological Space, if there exists a loop that cannot ...
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      1answer
      139 views

      Is this set of fractions dense in the interval $\big[\frac 13,\frac 12\big)$?

      I have an interest in the set $$A= \bigg\{\frac{ab+c}{(2a+1)b+c}\,\bigg|\, a \in {\mathbb Z}^+, b\in{\mathbb Z}^+~\text{is \((a+1)\)-smooth}, 0\leq c\leq ab\bigg\}.$$ In particular, is $A$ dense in ...
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      Is the set of escaping endpoints for $e^z-2$ completely metrizable?

      Let $f:\mathbb C \to \mathbb C$ be the complex exponential $$f(z)=e^z-2.$$ It is known that $J(f)$, the Julia set of $f$, is a uncountable collection of disjoint rays (one-to-one continuous images ...
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      32 views

      Sobolev tensor spaces and finite ranks

      Let $W^{2,2}(\Omega_i)$, $\Omega_i = [-1,1]$, $i = 1,\ldots,d$ be the Sobolev spaces of twice weakly differentiable, square integrable functions. Let further $\otimes_a$ denote the algebraic tensor ...
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      1answer
      316 views

      Normal bundle of Whitney embedding

      Let $X$ be a real $n$ dimensional manifold. One knows that it can be embedded into $\mathbb{R}^{2n}$ by the Whitney embedding theorem. The normal bundle for such an embedding will be a rank $n$ real ...
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      83 views

      Computing the number of topologies on a finite set [duplicate]

      Denote by $T(n)$ the number of non-homeomorphic topologies on a set with $n$ elements. I recently noticed that I am not aware of any good way of computing $T(n)$. Is there an interesting lower bound ...

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