Intereting Posts

Does the improper integral $\int_0^\infty\sin(x)\sin(x^2)\,\mathrm dx$ converge
How to divide using addition or subtraction
Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?
Probability of getting A to K on single scan of shuffled deck
How can I determine if three 3d vectors are creating a triangle
Characterization of Dirac Masses on $C(,\mathbb{R}^d)$
Find minimum of $a+b$ under the condition $\frac{m^2}{a^2}+\frac{n^2}{b^2}=1$ where $m,n$ are fixed arguments
A mouse leaping along the square tile
Finding XOR of all even numbers from n to m
Integer partition with fixed number of summands but without order
Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.
Proof that every number has at least one prime factor
Finding vector $x$ so that $Ax=b$ using Householder reflections.
Compactness of Algebraic Curves over $\mathbb C^2$
Definibility of $\mathbb{Z}$ in product rings

Let $f: X \to Y$ be continuous and proper (a map is proper iff the preimage of a compact set is compact). Furthermore, assume that $Y$ is locally compact and Hausdorff (there are various ways of defining local compactness in Hausdorff spaces, but let’s say this means each point $y \in Y$ has a local basis of compact neighborhoods).

Prove that $f$ is a closed map.

I know that this proof cannot require much more than a basic topological argument. But there’s just something that I’m missing.

- Is projection of a closed set $F\subseteq X\times Y$ always closed?
- Projection map being a closed map

We can start with $C \subseteq X$ closed, and then try to show that $Y \setminus F(C)$ is open (for each $q \in Y \setminus F(C)$, we would want to find an open set $V_q$ with $q \in V_q \subseteq Y \setminus F(C)$).

Hints or solutions are greatly appreciated.

- Compact spaces and closed sets (finite intersection property)
- A bounded net with a unique limit point must be convergent
- Prove the map has a fixed point
- Cover of (0,1) with no finite subcover & Open sets of compact function spaces
- Products and Stone-Čech compactification
- Show that $C_0(, \mathbb{R})$ is not $\sigma$-compact
- shortest path to Tychonoff?
- Closed sum of sets
- Theorem of Arzelà-Ascoli
- Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition

Let $C \subset X$ be closed. Let $y \in Y – f(C)$. Since $Y$ is locally compact, $y$ has a neighborhood $V$ with compact closure. Since $f$ is proper, $f^{-1}(\overline{V})$ is compact in $X$. Let $E = C \cap f^{-1}(\overline{V})$. $E$ is compact; thus, $f(E)$ is compact. Since $Y$ is Hausdorff, $f(E)$ is closed. Let $\hat V = V – f(E)$. $\hat V$ is a neighborhood of $y$ disjoint from $f(C)$ as desired.

- $\ell_\infty$ is a Grothendieck space
- What are the odds of cracking a cellphone pattern-lock?
- Universal closure of a formula
- How to generalize Reshetnikov's $\arg\,B\left(\frac{-5+8\,\sqrt{-11}}{27};\,\frac12,\frac13\right)=\frac\pi3$?
- Lebesgue's criterion for Riemann-integrability of Banach-space-valued functions?
- Maximum area of a square in a triangle
- Combinatorics/Task Dependency
- Proving that $X/R$ is Hausdorff $\implies$ $R$ closed.
- convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^{\lfloor \sqrt{n}\rfloor}}{\sqrt{n}}$
- Proving the Law of the Unconscious Statistician
- Finding the extrema of $E(\vec{r})=\frac1a x^2+\frac1b y^2+ \frac1c z^2$ with respect to constraints geometrically
- Evaluating this integral for different values of a constant
- Induction proof of $n^{(n+1) }> n(n+1)^{(n-1)}$
- How many of all cube's edges 3-colorings have exactly 4 edges for each color?
- Infinite dimensional integral inequality