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Results tagged with gn.general-topology
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user 1227
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
7
votes
Accepted
Potential connected non-Lie subgroup
I think the answer is, yes, the graph can be connected.
By definition, if the graph G is not connected, then we can find disjoint nonempty open sets A and B, such that G is contained in A union B. In …
- 6,571
11
votes
Foliation with leaves which are and are not dense
No, there doesn't exist such a foliation. The existence of any foliation would mean the Euler characteristic is zero, so the surface must be either a torus or a Klein bottle. Foliations for these surf …
- 6,571
6
votes
Dynamical properties of injective continuous functions on $\mathbb{R}^d$
This is an observation, along with a sketch of a potential construction where (1) does not imply (2) for $\mathbb{R}^3$.
Observation. I want to point out an example of a closed set $X \subset \mathbb …
- 6,571
5
votes
Accepted
Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynam...
No, this is usually not possible. There's a previous MO question discussing this, but to add to the material there:
A time-one map $\phi_1$ commutes with the 1-parameter family of diffeomorphisms $\p …
- 6,571
104
votes
Accepted
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Yes, the complement of any countable set in $\mathbb{R}^3$ is simply connected, by the Baire category theorem.
Say your set is $X = \{x_1, x_2, ... \}$, and let $y$ be any point in $\mathbb{R}^3 \set …
- 6,571