All Questions
Tagged with limits-colimits general-topology
86
questions
6
votes
1
answer
131
views
What is the subcategory of Top generated by the discrete spaces wrt limits and colimits?
In the category $\text{Top}$ of topological spaces, start with the subcategory $\text{Disc}$ of spaces equipped with the discrete topology (which is equivalent to $\text{Set}$). Then take its closure ...
- 432k
1
vote
1
answer
68
views
On the topology of $BO_k$
Let $BO_k$ be the classifying space given by:
$$BO_k=\varinjlim_{\mathbb N\ni n}Gr_k(\mathbb R^n)$$
I am trying to determine aspects about the topology of this space, but cannot find any sources that ...
- 3,431
2
votes
1
answer
79
views
Interpretation of closure in inverse limit
Can one interpret the closure of a set inside an inverse limit as the closure of its individual components? I have not been able to find a source confirming or denying this claim. I have only been ...
0
votes
1
answer
49
views
Colimits of full subdiagrams vs topological subspaces
Let $F:J \rightarrow \mathrm{Top}$ be a diagram in the category of topological spaces and $X:=\mathrm{colim} F$ be its colimit. For each $a \in \mathrm{ob}(J)$, denote by $\imath_a:F(a)\rightarrow X$ ...
- 151
3
votes
1
answer
142
views
Writing $\operatorname{Spec}\mathbb{Z}$ as an inverse limit of finite $T_0$-spaces
I want to show that $\operatorname{Spec}\mathbb{Z}$ can be written as an inverse limit of finite $T_0$-spaces. First off, $\operatorname{Spec}\mathbb{Z} = \{(0), (2),(3),(5),...\}$, so the closed sets ...
0
votes
0
answers
61
views
Show the category of Hausdorff spaces has Pushouts and Coequalizers [duplicate]
Let $\mathbf{Haus}$ be the category of Hausdorff spaces. I'm required to show that $\mathbf{Haus}$ has Pushouts and Coequalizers.
I don't know if I'm even close, but here is what I've tried:
I tried ...
- 821
2
votes
1
answer
90
views
Inclusion functor is final?
For $n\geq 1$ let $K_n$ be the partially ordered set of compact subsets of $\mathbb{R}^n$ ordered by inclusion. Let $B_n$ be the totally ordered set of closed balls in $\mathbb{R}^n$ centered at the ...
- 1,769
3
votes
1
answer
205
views
Profinite spaces are either finite or uncountable
I am reading about projective limits, and I am getting confused. In this answer, it is stated that the profinite spaces are either finite or uncountable. However, in the following example, I am ...
- 401
3
votes
0
answers
152
views
Are the limits of Euclidean spaces Hilbert spaces?
Consider each Euclidean space $\mathbb{R}^n$ as a topological vector space. I wondered what I get by taking $n \to \infty$ in the category of topological vector spaces over $\mathbb{R}$.
There are ...
- 2,049
3
votes
1
answer
164
views
A space homeomorphic to its own power
I sought for a topological space that is homeomorphic to its own power, e.g. $X \cong 2^X$ for the discrete two-point space $2$. Of course, $2^X$ has the compact-open topology.
Here's my approach: ...
- 2,049
1
vote
0
answers
38
views
Is the long line a colimit in a suitable category, if not in $\mathbf{Top}$?
The only definitions of the so-called long line I've encountered so far all consist, with only slight variations, in forming a long ray by means of the lexicographical order on $\omega_1\times[0,1)$, ...
- 697
1
vote
1
answer
210
views
Is inverse limit topology closed in product topology?
Suppose $\{X_i\}_{i\in I}$ is an inverse system of topological spaces. Denote $X=\varprojlim X_i$. It's naturally a subspace of the product topology $\prod_{i} X_i$. I want to ask: is $X$ closed in ...
- 1,424
3
votes
0
answers
89
views
Topology determined by cubes
Which spaces $X$ have the property that $X \to Y$ is continuous if and only if $I^n \to X \to Y$ is continuous for all $I^n \to X$?
Example: manifolds and CW complexes have this property, since we ...
- 5,994
2
votes
1
answer
135
views
Can the topological spaces $\mathbb{R}$ and $\mathbb{Q}$ be expressed as a colimit of a diagram of discrete/finite spaces?
I am interested in knowing if $\mathbb{R}$ and $\mathbb{Q}$ are colimits in the category of topological spaces of a diagram $J$ of discrete or finite spaces. I would like to know also if it is ...
- 23
4
votes
1
answer
69
views
Projective Limits of Compact Groups: Exact or Not?
I am reading the following lemma from Washington's book "Introduction to Cyclotomic Fields":
On the other hand, there is a counterexample, given by this answer. The comments below this ...
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