Intereting Posts

Example of non-Noetherian non-UFD Krull domain?
Are all algebraic integers with absolute value 1 roots of unity?
Limit $\lim_{n\to\infty}\sqrt n(\sqrtx-1)$
A $\log \Gamma $ identity: Where does it come from?
If $(x_n) \to x$ then $(\sqrt{x_1x_2\cdots x_n}) \to x$
Series $\sum\limits_{n=1}^\infty \frac{1}{\cosh(\pi n)}= \frac{1}{2} \left(\frac{\sqrt{\pi}}{\Gamma^2 \left( \frac{3}{4}\right)}-1\right)$
Derivatives of the commutator of flows (or, what are those higher derivatives doing in my tangent space?!)
measure theory question about densities
What exactly is a tensor product?
How do you go about doing mathematics on a day to day basis?
Solve the congruence system $\ x\equiv m_i-1 \pmod{m_i}\,$ for $\,i = 1,\ldots, k$
Algebraic way to change limits of integration of a double integral
Is There a Problem with This Professor's Proof Concerning Interior and Boundary Points?
How to prove $\sum_{s=0}^{m}{2s\choose s}{s\choose m-s}\frac{(-1)^s}{s+1}=(-1)^m$?
Find number of integral solutions of $\sqrt{n}+\sqrt{n+7259}$

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, then $\times X$ has a right adjoint, namely $Y\mapsto Y^X$(equipped with compact-open topology). I also heard that the right adjoint of $\times X$ doesn’t exist for every topological space $X$. So I wonder is there any more visible description of the class of topological spaces $X$ such that the left adjoints of$\times X$ exist? Any idea will be appreciated.

- Grp as a reflexive/coreflexive subcategory of Mon
- What is the way to see $(S^1\times S^1)/(S^1\vee S^1)\simeq S^2$?
- About maps between homology groups
- Homotopy functions
- When is a fibration a fiber bundle?
- Non-orientable 1-dimensional (non-hausdorff) manifold
- How much is cohomotopy dual to homotopy?
- Are these two spaces homotopy equivalent?
- Fundamental group of $S^2$ with north and south pole identified
- Original Papers on Singular Homology/Cohomology.

In general, in a category $\mathsf{C}$ that has products, an object $c$ such that $- \times c : \mathsf{C} \to \mathsf{C}$ has a right adjoint is called **exponentiable**. There are various characterizations of exponentiable objects in $\mathsf{Top}$, and you can find some of them in this $n$Lab article:

Theorem (Exponentiability, I).An object $X$ of $\mathsf{Top}$ is exponentiable if and only if $X \times -: \mathsf{Top} \to \mathsf{Top}$ preserves coequalizers, or equivalently quotient spaces.For open subsets $U$ and $V$ of a topological space $X$, we write $V \ll U$ to mean that any open cover of $U$ admits a finite subcover of $V$; this is read as $V$ is relatively compact under $U$ or $V$ is way below $U$. We say that $X$ is

core-compactif for every open neighborhood $U$ of a point $x$, there exists an open neighborhood $V$ of $x$ with $V \ll U$. In other words, $X$ is core-compact iff for all open subsets $V$, we have $V=\bigcup \{ U \mid U \ll V \}$.

If $X$ is Hausdorff, then core-compactness is equivalent to local compactness.

Theorem (Exponentiability, II).An object $X$ of $\mathsf{Top}$ is exponentiable if and only if it is core-compact.

In particular, a Hausdorff space is exponentiable iff it is locally compact.

A point of warning: in general, even if $Y^X$ exists, it may be the case that $Y^X$ is not exponentiable itself! This has lead to a search for what is called a convenient category of topological spaces: you can think of it as a category of spaces (a full subcategory of $\mathsf{Top}$) that you can use for all your usual operations. There are multiple candidates, for example compactly generated weakly Hausdorff spaces. It all depends on what you want to do with them.

In answer to a question that came up in comments under Najib’s answer, let me point out that the category of pseudotopological spaces is a locally small cartesian closed category that contains $\text{Top}$ as a full subcategory. Actually, this category $\text{PsTop}$ has the even stronger property of being complete and locally cartesian closed and indeed is a quasitopos.

Exactly the same remarks apply to the even larger category of convergence spaces (larger in the sense that pseudotopological spaces are special types of convergence spaces).

Moreover, we have the very nice fact that the inclusion functor $i: \text{Top} \hookrightarrow \text{PsTop}$ preserves exponential objects, in the sense that if $X, Y$ are topological spaces such that the exponential $Y^X$ exists in $\text{Top}$, then the canonical map $i(Y^X) \to i(Y)^{i(X)}$ is an isomorphism (of course, the exponential on the right is formed *qua* the cartesian closed structure on $\text{PsTop}$).

One way of thinking about this is that for any topological spaces $Y, X$, the natural candidate for an exponential topology on the set of continuous functions $\hom_{\text{Top}}(X, Y)$ (if one exists) is always given by *continuous convergence*, meaning the topology that is naturally associated with the convergence relation that defines the exponential in $\text{PsTop}$. That is, a topology (called the *natural topology* on $\hom_{\text{Top}}(X, Y)$) can always be formed that corresponds to continuous convergence, and so must be the exponential topology on $Y^X$ whenever there is one, and it is always the case that this natural topology is the finest topology on $\hom_{\text{Top}}(X, Y)$ such that given any topological space $A$ and a continuous function $A \times X \to Y$, the function $A \to \hom_{\text{Top}}(X, Y)$ formed by currying is also continuous. For basics on continuous convergence, see these notes by Mike Shulman (and the references given therein).

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- proving $\sum\limits_{k=1}^{n} \Bigl\lfloor{\frac{k}{a}\Bigr\rfloor} =\Bigl\lfloor{\frac{(2n+b)^{2}}{8a}\Bigr\rfloor} $
- Let $f(x) =7x^{32}+5x^{22}+3x^{12}+x^2$. Then find its remainder in the following cases.
- Prove weak convergence of a sequence of discrete random variables
- Difference between variables, parameters and constants
- How to integrate $\int_0^\infty\frac{x^{1/3}dx}{1+x^2}$?