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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{ …

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: $ …

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 …

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 …

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. …

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 $[0,1]$ …

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 …

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 …

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 …

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 …

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 …

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) …

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 …

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 …