All Questions
Tagged with limits-and-colimits gn.general-topology
21 questions
1
vote
0
answers
127
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Extremally disconnected sets as building blocks for compact Hausdorff spaces
Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
- 1,638
6
votes
1
answer
261
views
When does base-change in topological spaces preserve quotient maps?
The question when $(-) \times X$ preserves colimits in topological spaces is well-studied. Since it always preserves arbitrary coproducts (disjoint unions), one only has to show when it preserves ...
- 12.1k
9
votes
1
answer
734
views
Does the category of locally compact Hausdorff spaces with proper maps have products?
nlab presents a proof that the category of locally compact Hausdorff spaces does not admit infinite products in general. In particular it shows that there is no infinite product of $\mathbb{R}$, since ...
- 295
14
votes
2
answers
761
views
Is there a large colimit-sketch for topological spaces?
Question. Is there a large colimit-sketch $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$?
In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}...
- 63.1k
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 ...
- 5,405
0
votes
1
answer
81
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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
$$
C_{\omega_i}(\mathbb{R}^n,\...
- 5,405
1
vote
1
answer
379
views
Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
- 5,405
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 $X$ ...
- 5,405
1
vote
1
answer
303
views
Comparison of product topology and colimit topology in sequence spaces
In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by:
$$
d(x,y)\triangleq \sup_{n \in \mathbb{N}} d(x_n,y_n)
$$
is strictly finer than the product topology on $\prod_{n \...
- 5,405
1
vote
0
answers
222
views
Surjectivity of colimit maps for topological spaces
From this post and to (co)completness of the category Top of topological spaces and continuous functions we know that for any diagram $B_i$ and an object $A$ in Top, there are natural maps of sets:$\...
- 5,405
5
votes
1
answer
698
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 ...
- 5,405
0
votes
1
answer
203
views
Filtered colimit of a topological space
Suppose that $X$ is a space filtered by closed subspaces $X_{1}\subset X_{2}\subset \dots$.
As topological space $X=\operatorname{colim}_{n}X_{n}$.
We define $Y_{n}=X_{n+1}/X_{n}$, and consider the ...
- 511
1
vote
0
answers
131
views
When is a nested sequence of closed sets a colimit?
Let $X$ denote a topological space and $X_0\subset X_1\subset \ldots\subset X$ a nested sequence of closed subsets of $X$ such that $$ \bigcup_i X_i =X$$
It is easy to see that in the general case $X$...
- 1,174
2
votes
3
answers
235
views
Example of an $\omega_1$ decreasing chain of dense semicontinua?
In his well-known paper Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:
We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \...
- 1,955
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 ...
- 1,270
8
votes
0
answers
291
views
Loop space functor and sequential colimits of inclusions
The question is about a fact that is mentioned as "evident" everywhere in the literature, so my guess is that some small detail is passing over my head. Here it is:
Let $X_0\hookrightarrow X_1 \...
- 91
7
votes
1
answer
2k
views
Direct limit of compact topological spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a direct system of compact topological spaces, meaning that we have morphisms $f_i\colon X_i \to X_{i+1}$ with the necessary compatibility conditions. Is there any ...
- 2,384
3
votes
0
answers
867
views
The inductive and projective limits of compact Hausdorff topological groups
Are there conditions known under which the inductive or projective limit of a family of compact Hausdorff topological groups is compact? (For instance, such a result is valid for the projective limit ...
- 5,407
2
votes
0
answers
562
views
Direct Limits and Limits of Nets
A net is a function from a directed set into a topological space, and it is said to converge to a point if certain conditions are satisfied. Similarly, a direct system is a function from a directed ...
- 15.4k
20
votes
2
answers
2k
views
Is every compact topological ring a profinite ring?
There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
- 2,200
17
votes
10
answers
3k
views
References for homotopy colimit
(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
- 12.8k