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

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31 views

### Connected and homogeneous $T_2$-space not homeomorphic to a subset of $\mathbb{R}^n$

What is an example of an connected and homogeneous $T_2$-space $(X,\tau)$ with $|X|=2^{\aleph_0}$ such that for no $n\in\mathbb{R}$ the space $(X,\tau)$ is homeomorphic to a subspace of $\mathbb{R}^n$?...

**0**

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48 views

### How to prove that the set of the diffeomorphisms is open set in the space $C^1$ [on hold]

How to prove that the set of the diffeomorphisms is open set in the space $C^1$? Can anyone indicate a book, or give some suggestion?

**5**

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87 views

### What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales:
...the single most important fact which distinguishes locales from spaces: the ...

**4**

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**1**answer

113 views

### Is the Euclidean topology on $\mathbb{R}$ contained in a maximal connected topology?

If $(X,\tau)$ is a connected space, then $\tau$ need not be contained in a maximal connected topology.
Is the Euclidean topology on $\mathbb{R}$ contained in a maximal connected topology?

**0**

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88 views

### CW-complex structure on the quotient

Let $X$ be an $n$-dimensional CW-complex and let $A \subseteq X$ be a subcomplex. I want to show that the quotient space $X/A$ admits a structure of a CW-complex with skeletons $(X/A)^j := \pi(X^j \...

**1**

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**1**answer

82 views

### The preimage of continuum on Torus

Let $p:\mathbb{R}^2\rightarrow\mathbb{R}^2/\mathbb{Z}^2$ be the natural projection, obviously $\mathbb{R}^2/\mathbb{Z}^2$ is the torus $\mathbb{T}^2$, if $K$ is a connected and compact subset of $\...

**4**

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**1**answer

91 views

### $T_2$-spaces in which no two open sets are homeomorphic

This question was about spaces in which all non-empty open sets "look alike".
Now I am interested in the opposite: Is there a $T_2$-space $(X,\tau)$ with $|X|>1$ such that whenever $U\neq V$ are ...

**1**

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**1**answer

58 views

### Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...

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**1**answer

129 views

### effectively distinguishing knots

It was proven, I think by Mijatovi? EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound,...

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63 views

### A community effort: equilibrium in quitting games

This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...

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**1**answer

41 views

### Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

This is a cross-post to the question I asked at MSE.
The set of Schur stable matrices is
\begin{align*}
\mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\},
\end{align*}
where $\rho(\cdot)$ denotes ...

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52 views

### Topological Shape Operator More Sensitive than Inverse Limits

This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...

**3**

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**1**answer

62 views

### sequences of iterated orthogonals (lifting property) in a category

I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property.
For example, several iterated orthogonals of $ \emptyset\...

**2**

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**1**answer

72 views

### Is the topology generated by the union of a chain of paracompact topologies paracompact?

Let $X$ be a set and let ${\frak T}$ be a collection of paracompact topologies on $X$ such that for any $\tau, \tau'\in {\frak T}$ we have $\tau\subseteq \tau'$ or $\tau'\subseteq \tau$. Let $\sigma$ ...

**1**

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**1**answer

149 views

### Is every paracompact topology contained in a maximal paracompact topology?

If $(X,\tau)$ is a paracompact, is there a topology $\tau'\supseteq \tau$ such that $(X,\tau')$ is still paracompact, and $\tau'$ is maximal with respect to $\subseteq$ and paracompactness?