A topological space $(X,\mathcal T)$ is said to be divisible iff for each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a symmetric neighborhood $V$ of the diagonal such that: $$V\circ V\subseteq U$$ Is every $T_4$ topological space divisible?

I have some problems understanding the proof of the following lemma: Lemma: Let $x \in X, \ \ \ U, W \in \mathcal{U}, \ \ \ \mathcal{T(U)}$ is the topology on $X$. If there exists $V \in \mathcal{U}$ such that $U \circ V \subset W$, then $U[x] \subset intW[x]$ and $\overline{U[x]} \subset W[x]$. $\mathcal{U}$ is […]

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily that: There’s some neighborhood $U$ of $1$, the identity element of the group $G$, such that $\overline{UU}$ is […]

If X is any uniformizable topological space, then we can show that there exists a finest covering uniformity on X by taking the union of all covering uniformities compatible with the topology of X. We know that this finest uniformity is a normal family. How to show that this finest uniformity is compatible with the […]

Given a uniformizable (w.r.t. entourage uniformity) space $X$ there is a finest uniformity on $X$ compatible with the topology of $X$ called the fine uniformity or universal uniformity. A uniform space is said to be fine if it has the fine uniformity generated by its uniform topology. why the fine uniformity is exists and How […]

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise to a uniform structure via: $$\Phi:=\uparrow\{U_N:N\in\mathcal{N}_0\}\text{ with }U_N:=\{(x,y):y-x\in N\}$$ where $C\in\uparrow\mathcal{A}$ iff $A\subseteq C$ for some $A\in\mathcal{A}$. Moreover prove that the uniform topology coincides […]

I work on reloids, a generalization of uniform spaces. As such I am interested about properties of totally bounded uniform spaces. My question: What are interesting properties of totally bounded uniform spaces? (The more theorems/propositions you give, the better.)

I’ve seen in various textbooks and notes that if $X$ is paracompact, then the collection of all the neighborhoods of the diagonal is a uniformity. I am trying to show that this uniformity is complete using Cauchy filters. So far, I let $\mathfrak{F}$ be a filter on $X$ that does not converge. By definition, saying […]

Let $X$ and $Y$ is a uniform spaces. Let $f$ is a uniformly continuous surjective function $X\rightarrow Y$. Conjecture: If $X$ is totally bounded then $Y$ is also totally bounded.

The long line $L$ is uniformizable; in fact, as $L$ is a locally compact Hausdorff space we can explicitly write down a uniformity for it: If $\hat{L}$ is the one-point compactification of $L$, then $\hat{L}$ is compact Hausdorff, and so it has a natural uniformity that we can restrict to $L$. But this is not […]

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