# Questions tagged [gn.general-topology]

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

2,953 questions
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### Non embedding of $Y\times Y$ into $\mathbb{R}^3$

I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type. Theorem. Let $Y$ be the union of two segments ...
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### Is each cometrizable space a subspace of a cometrizable topological group?

Following Gruenhage we call a topological space $X$ cometrizable if $X$ admits a weaker metrizable topology such that every point $x\in X$ has a (not necessarily open) neighborhood base consisting of ...
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### Prove that $\mu \left(\left\{t\in X\,;\;\sum_{i=1}^d|\phi_i(t)|^2>r \right\}\right)=0$

Let $(X,\mu)$ be a measure space and $\phi=(\phi_1,\cdots,\phi_d)\in L^{\infty}(X)$. Let $$r=\max\left\{\sum_{i=1}^d|z_i|^2; (z_1,\cdots,z_d)\in \mathcal{C}(\phi)\right\},$$ where $\mathcal{C}(\phi)$...
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### Lorenz attractor path-connected?

Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure. EDIT: The answer below is unsatisfactory, and possibly ...
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### The dominating number $\mathfrak{d}$ and convergent sequences

All spaces considered below are compact Hausdorff. If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...
### What is the boundary of the set $\{ x : dist (x ,\partial \Omega) > \alpha \}$ for a domain $\Omega$?
Let $\Omega$ is a bounded open domain in $\mathbb R ^n$, and $\alpha \geq 0$ a real number, and consider the set $E_\alpha = \{ x \in \Omega : \text{dist}(x , \partial \Omega) > \alpha\}$, which ...