All Questions
Tagged with cartesian-closed-categories gn.general-topology
6
questions
15
votes
3
answers
1k
views
Why it is convenient to be cartesian closed for a category of spaces?
In 1967 Steenrod wrote what later became a quite celebrated paper, A convenient category of topological spaces (Michigan Math. J. 14 (1967) 133–152). The paper conveys the work of many (among the most ...
- 8,027
4
votes
0
answers
68
views
Need to know if a certain full subcategory of Top is cartesian closed
Consider the full subcategory of Top consisting of all spaces $X$ such that a subset $A$ of $X$ is closed if and only if $A \cap K$ is closed in $K$ for all subspaces $K$ of $X$ which are countably ...
- 1,943
7
votes
2
answers
255
views
The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$
Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the ...
- 6,641
3
votes
1
answer
222
views
Is the category of convergence spaces cartesian-closed?
Convergence spaces are a generalization of topological spaces; we denote the category of convergence spaces with continuous maps with ${\bf Conv}$. Is ${\bf Conv}$ cartesian-closed?
- 44.5k
29
votes
1
answer
2k
views
Is there a topological space X homeomorphic to the space of continuous functions from X to [0, 1]?
In general, we might ask when we can find interesting spaces $X, Y$ such that $X$ is homeomorphic to $[X, Y]$. By the Lawvere fixed point theorem $Y$ must have the fixed point property. Happily, $Y = [...
- 114k
8
votes
2
answers
418
views
Is the category of quotient of countably based topological spaces cartesian closed ?
In "Handbook of categorical algebra Vol 2" from Francis Borceux, the author gives a proof that $Top$ is not cartesian closed. It seems to me that this proof can be adapted to show that the category $\...
- 774