Intereting Posts

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$
The Continuum Hypothesis & The Axiom of Choice
Sine function dense in $$
Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class
Finding the shortest distance between two lines
Does such a finitely additive function exist?
Tangent space in a point and First Ext group
Why isn't there a contravariant derivative? (Or why are all derivatives covariant?)
How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?
Sum of discrete and continuous random variables with uniform distribution
Terry Tao, Russells Paradox, definition of a set
Prove these two segments are equal
Testing whether an element of a tensor product of modules is zero
Derivative of convolution
Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$

Let $X$ be a topological space. The endofunctor $\_\times X$ of the category of all topological spaces does in general not possess a right adjoint, since the category is not cartesian closed.

- Is it nevertheless true in general, that $colim(A_i)\times X\cong colim(A_i\times X)$?
- If 1) is false: Is it at least true in full generality for colimits indexed over the natural numbers $…\rightarrow X_i\rightarrow X_{i+1}\rightarrow …$?
- If 2) is false: What are topological conditions on $X_i$ and $A$, such that 2) gets right?

- Why does Seifert-Van Kampen not hold with $n$-th homotopy groups?
- An inductive construct of the Hausdorff reflection
- Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell
- Laymans explanation of the relation between QFT and knot theory
- What's the dual of a binary operation?
- The projective model structure on chain complexes
- Calculate the Wu class from the Stiefel-Whitney class
- Real projective plane: $f_*$ isomorphism $\implies f$ surjective
- Co/counter variancy of the Yoneda functor
- Category Theory usage in Algebraic Topology

Here is a counterexample for the directed limit:

Let $X_n$ be a wedge of $n$ circles $C_1,\dots,C_n$. We have a sequence of closed inclusions $X_1\hookrightarrow X_2\hookrightarrow\dots$ whose colimit is $X$, a wedge of countably many circles. There is a continuous bijection $j:\text{colim}(X_n\times\Bbb Q)\to X\times\Bbb Q$ which I claim is not a homeomorphism. Namely, if $A$ is the subset of $\text{colim}(X_n\times\Bbb Q)$ whose intersection with the cylinder $C_n\times\Bbb Q$ is

$$

\left\{ (e^{2\pi i x},y)\ \middle|

\ \frac\pi n \le y \le \frac\pi n +\min(x,1-x)\right\}

$$

then $A$ is closed, but $j(A)$ is not closed in $X\times\Bbb Q$ as $(0,0)$ is a limit point of $j(A)$.

We would have a homeomorphism, though, if $Y$ (which in the example was $\Bbb Q$) were locally compact, as that would make the functor $(-)\times Y$ a left adjoint to the functor $(-)^Y$, and left adjoints preserve all colimits.

It also works, for any $Y$, if the diagram is such that we can choose a subset $\cal S$ of the spaces in the diagram $X:\mathscr J\to\mathbf{Top}$ such that

- every space in the diagram is either in $\cal S$ or has a map to some space in $\cal S$
- the quotient map $\coprod_{i\in\mathscr J} X_i\to\text{colim}(X_i)$ restricts to a perfect map $\coprod_{s\in\mathcal S}X_s\to\text{colim}(X_i)$ (Note that the previous point implies that this restriction is a quotient map).

This is because for every perfect map $p:X\to Y$, the product $p\times\mathbf 1_Z:X\times Z\to Y\times Z$ is closed, thus a quotient map.

- Products of homology groups
- Infinitely many primes of the form $8n+1$
- Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$
- Memorizing the unit circle?
- fundamental groups of open subsets of the plane
- Automorphism of the Field of rational functions
- Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.
- Equality of mixed partial derivatives
- Where do bitopological spaces naturally occur? Do they have applications?
- Calculating probability with n choose k
- Find the inverse of a Trig Matrix
- A family having 4 children has 3 girl children. What is the probability that their 4th child is a son?
- Tensor products of fields
- Explain these ring isomorphisms
- Convergence of $f\in L_1(a,b)$ and a sequence of step functions