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

73 votes
Accepted

In a topological space if there exists a loop that cannot be contracted to a point does ther...

Here is an example of topological space $X$, embeddable as compact subspace of $\mathbf{R}^3$, that is not simply connected, but in which every simple loop is homotopic to a constant loop. Namely, st …
YCor's user avatar
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25 votes
Accepted

Is there a differentiable map surjective from low to high dimension?

$\DeclareMathOperator\R{\mathbf{R}}$It's easy to check that the image of any locally Lipschitz map $f:\R^n\to\R^m$ has measure zero when $n<m$ (this encompasses the case of class-$\text{C}^1$ maps, bu …
YCor's user avatar
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24 votes

Elementary proof that $\mathbb{R}^3 \setminus \{p_1,\dots,p_n\}$ is not homeomorphic to $\ma...

Let $P_n$ be the property for a Hausdorff topological space $X$: for every compact subset $K$ of $X$, there exists a compact subset $L$ of $X$ such that $K\subset L$ and $X-L$ has exactly $n$ componen …
YCor's user avatar
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23 votes

A question about subsets of plane

(Initial post November 24, 2016, edited November 27, 2016) This does not exist. The proof that $X$ doesn't exist is a bit elaborate and makes use of ends of coset spaces. I will prove: (a) Let $\ …
YCor's user avatar
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20 votes
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Existence of infinite groups that are too reluctant to be topological

There is a large literature about this, see "non-topologizable groups". These are, by definition, groups for which the only Hausdorff group topology is discrete. There are various examples, the first …
YCor's user avatar
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19 votes
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Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?

No. First observe that the automorphisms of the semigroup $\mathbf{N}^*$ (which you denote $\mathbb{N}$) are induced by permutations of primes. Consider the automorphism $f$ induced by the transposi …
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18 votes
Accepted

Is there a compact, connected, totally path-disconnected topological group?

(I'm assuming the groups to be Hausdorff to avoid the discussion degenerate into idle banter.) The answer is yes: $\{1\}$ is such a group. The answer to the intended question (which is probably whet …
YCor's user avatar
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14 votes
Accepted

Do all homogeneous spaces have homogeneous compactifications?

Since you want a connected example: A surface of infinite genus has no homogeneous compactification. Indeed first observe a dense locally compact subset has to be open. So the surface has to be open, …
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14 votes

Connected space being not locally connected at each point

There are compact groups that are connected, but not locally connected at any point. For instance, solenoids $(\mathbf{R}\times\mathbf{Z}_p)/\langle (1,1)\rangle$. (It is locally homeomorphic to $\mat …
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13 votes
Accepted

Must a path of compact sets in $X$ descend to a path in $X$?

$\DeclareMathOperator{\R}{\mathbf{R}}\DeclareMathOperator{\Z}{\mathbf{Z}}$The answer is no, even in the circle (and hence in the plane). As coordinates, write the circle as the 1-point compactificatio …
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12 votes
Accepted

About locally compact groups without compact subgroups

Yes, it's even a Lie group whose unit component is a semidirect product $R\rtimes S^n$, where $R$ is a simply connected solvable Lie group and $S$ is the universal covering of $\mathrm{SL}_2(\mathbf{R …
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11 votes
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Connecting a compact subset by a simple curve

Not always. Let $K$ be a subset of an ambient space $V$ ($V=\mathbf{R}^2$ is fine, but doesn't matter) that is the closure of a discrete subset $D$, such that $K-D$ is homeomorphic to a segment. This …
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11 votes
Accepted

Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)?

There's no topologically mixing self-homeomorphism of the circle. Indeed, pick 3 points, so that the complement of these 3 points consists of 3 intervals $A,B,C$. If $g$ is a self-homeomorphism such t …
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11 votes
Accepted

Topological group locally homeomorphic to the Hilbert cube

The answer is no. Since the Hilbert cube is compact and locally contractible, such a group would be a locally contractible locally compact group. And every locally contractible locally compact group i …
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11 votes
Accepted

Non-homeomorphic connected one-dimensional Hausdorff spaces that have continuous bijections ...

Here's such a construction, actually producing an infinite family of such spaces, actually planar and locally compact, pairwise in continuous bijection in both directions but pairwise non-homeomorphic …
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