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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
3
votes
1
answer
378
views
Closed embedding into a normal Hausdorff space and left lifting property
I am trying to understand the characterization of the class of closed embeddings into a normal Hausdorff space as the class of continuous maps satisfying the left lifting property with respect to a un …
2
votes
0
answers
92
views
Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-co …
2
votes
0
answers
61
views
Space of continuous paths up to strict reparametrization
Take a Hausdorff topological space $X$. Take two distinct points $x$ and $y$ of $X$. Consider a set $U$ of continuous paths $p$ from $[0,1]$ to $X$ equipped with the compact-open topology such that: $ …
8
votes
1
answer
275
views
Compact-open topology and Delta-generated spaces
Consider the set of continuous maps $C^0([0,1],[0,1])$ equipped with the compact-open topology. It is metrisable, and therefore sequential. It is also a k-space: see http://neil-strickland.staff.shef. …
2
votes
2
answers
252
views
Continuous bijection between two homotopy equivalent $\Delta$-generated spaces
EDIT:
First edit after an interesting answer.
$(S,\mathcal{T}_1)$ and $(S,\mathcal{T}_2)$ are homotopy equivalent to the same Quillen cofibrant space.
Let $S$ be a set with two topologies $\mathcal{ …
3
votes
0
answers
331
views
About the Moore composition of paths
1) QUESTION (EDIT: 04/28/2020 to remove a possible counterexample)
I work with weak Hausdorff $k$-spaces (so all spaces are $T_1$). The internal hom is denoted by $\mathbf{TOP}(-,-)$. Let $\mathcal{G …
3
votes
0
answers
132
views
Colimits of weak Hausdorff $k$-spaces
Notations:
$\mathbf{T}$ is the category of weak Hausdorf $k$-spaces.
$\mathbf{K}$ is the category of $k$-spaces.
Fact: The inclusion functor $\mathbf{T} \subset \mathbf{K}$ is a right adjoint. It …
1
vote
0
answers
138
views
Identifying the two points of a subspace homeomorphic to a Sierpinski space
Let $X$ be a $\Delta$-generated space having a subset $A=\{a,b\}$ such that the relative topology is the Sierpinski topology with for example $\{a\}$ closed and $\{b\}$ open (the Sierpinsky space is a …
2
votes
Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom
Concerning your third question, the cogenerators of the category of general topological spaces are precisely the non-$T_0$-spaces. See Example 7.18 Remark (4) in Adamek, Herrlich and Strecker's Abstra …
8
votes
What was Burroni's sketch for topological spaces?
The category of topological spaces is the category of models of a relational universal strict Horn theory $T$ without equality, i.e. the axioms are of the form $(\forall x)(\phi(x) \rightarrow \psi(x) …
0
votes
Which sequential colimits commute with pullbacks in the category of topological spaces?
It is not an answer and I cannot let something wrong: the isomorphism is a general fact about locally finitely presentable categories. Let $\mathcal{K}$ be a locally presentable category. The pullback …
7
votes
Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop ...
I don't know for Peter May's work. I know that the weak Hausdorff condition can be useful for the following reason: let $f:X\to Y$ be a continuous map between Hausdorff k-spaces. Then consider the equ …
3
votes
Categorical Construction of Quotient Topology?
Nobody gave this reference so I give it : http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html, "The joy of cats", especially chapter 21 p350. The notions of initial and final topologies are ge …
3
votes
1
answer
203
views
Topological question about right-lifting property and the evaluation map
Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ …