Skip to main content

All Questions

Filter by
Sorted by
Tagged with
1 vote
1 answer
379 views

Creating an inverse system which "stratifies density"

Setting: Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying $$ \bigcup_{n ...
1 vote
0 answers
81 views

Examples of spaces which have explicit expression as colimits in $\mathrm{Top}$

$\DeclareMathOperator\Ball{Ball}$Question: What "well-known" spaces can be explicitly written down in the form $\bigcup_k \phi_k C(K_n,\mathbb{R}^m)$; where $K_n$ is a non-empty compact ...
0 votes
1 answer
81 views

Ultrabornological representation for the space of uniformly continuous functions?

Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space $$ C_{\omega_i}(\mathbb{R}^n,\...
1 vote
1 answer
511 views

Convergence in $C_c$ but not in $C$

Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ ...
5 votes
1 answer
699 views

Can $L^1_{loc}$ be represented as colimit?

Let $L^1_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1_{m_K}$ denote the space ...
5 votes
0 answers
211 views

A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff

We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...