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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 …
David Roberts's user avatar
  • 35.5k
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) …
Philippe Gaucher's user avatar
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 …
Philippe Gaucher's user avatar
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 …
Philippe Gaucher's user avatar
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 …
Philippe Gaucher's user avatar
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 $[0,1]$ …