# Questions tagged [gn.general-topology]

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

4,052 questions
Filter by
Sorted by
Tagged with
34 views

### Topological rings with a final topology

Given a family of ring homomorphisms $\phi_i : X \rightarrow Y_i$ where each $Y_i$ is a topological ring and consider the initial topology on $X$, i.e. the coarest topology such that each map is ...
• 151
40 views

• 5,057
43 views

### Can we decompose an increasing net of functions into two increasing nets with prescribed supports?

Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...
• 5,057
49 views

### Continuity of "inversion operator" between function spaces

Question: When is the operation of inversion continuous as a map between spaces of invertible functions? Let $\mathcal{F}$ be a function space such that $f\in\mathcal{F}\implies$ $f$ is invertible and ...
37 views

• 10.6k
56 views

• 51.1k
308 views

### How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?

I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere. Here $\Omega$ is any ...
• 865
42 views

### Properties on morphism of locally convex vector spaces

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $U,V,W,W'$ $K$-vector spaces, such that $U$ is a Banach-space and $W,W'$ are finite dimensional. Further we have an (algebraic) short exact ...
• 911
81 views

### Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?

Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
• 10.6k
147 views

• 1,364
124 views

### Subsets of $\mathbb{S}^n$ fixed by an orientation-reversing self-homeomorphism — Part 1

Call a subset $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing self-homeomorphism $h: \mathbb{S}^n \to \mathbb{S}^n$ fixing $Z$ pointwise. Question 1: Which subsets of ...
• 1,364
150 views

74 views

### Is there a bound on the number of connected components of a zero set of an integrable function?

If $f$ is a real-analytic function on $[0,1]^n$, and $f$ has finite differential transcendence degree, is there some way to bound the number of connected components of its zero set or the set where it ...
179 views

### Steinhaus number of a group

$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$. Let $\mathcal A_X$ be the family of ...
• 34.8k
1 vote
Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism. Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...