# Questions tagged [gn.general-topology]

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

3,031
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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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46 views

### 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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86 views

### 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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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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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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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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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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280 views

### 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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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 ...