Intereting Posts

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Why is sum of a sequence $\displaystyle s_n = \frac{n}{2}(a_1+a_n)$?
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Hard Olympiad Inequality
Must an ideal contain the kernel for its image to be an ideal?
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How to show $\alpha_d : M_d \to \Gamma(X, \widetilde{M(d)})$ an isomorphism for sufficiently large $d$?
Erdős and the limiting ratio of consecutive prime numbers
Why are Hornsat, 3sat and 2sat not equivalent?

Suppose $f:X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Are the following results true?

$1$. The map $f$ takes discrete sets to discrete sets.

$2$. If $f$ is injective, then $f$ must be a homeomorphism onto its image.

- Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.
- Compactness in the weak* topology
- Why is compactness so important?
- A and B disjoint, A compact, and B closed implies there is positive distance between both sets
- About $ \{ x \in^{\omega_1}:|\{\alpha<\omega_{1} :x(\alpha)\ne 0 \}|\le\omega \}$
- Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ , then $f$ is isometry?

**Edit**1:Let $A$ be discrete in $X$ and let $K$ be compact in $Y$ then $f(A) \cap K=f(A \cap f^{-1}(K))$,is finite since $A \cap f^{-1}(K)$ is finite.Hence $f(A)$ is discrete.

As there is a counterexample in answer below so can someone please point out the error in my proof?

- What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$?
- Construct a metric for the topology of compact convergence on $Y^{X}$ so that $Y^{X}$ is complete when $(Y,d)$ is complete and $X$ is $\sigma$-compact
- What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?
- Existence of exhaustion by compact sets
- Prove that the expression is a perfect square
- If $f \circ g$ is onto then $f$ is onto and if $f \circ g$ is one-to-one then $g$ is one-to-one
- when is rational function regular?
- Is there a name for the operation $f^{-1}(f(x) \oplus f(y))$?
- How to show that this set is compact in $\ell^2$
- Inverse function of $y=\frac{\ln(x+1)}{\ln x}$

Observe that a map $f:X\to Y$ to a compactly generated space $Y$ is a closed map if for every compact set $K$ of $Y$, the restriction $f_K:f^{-1}(K)\to K$ is closed. For if $C$ is closed in $X$, then $f(C)\cap K=f(C\cap f^{-1}(K))=f_K(C)$ is closed in $K$, hence $f(C)$ is closed in $Y$.

As a consequence, a proper map $f:X\to Y$ to a compactly generated Hausdorff space is closed. That’s because if $K$ is compact in $Y$, then $f_K$ is a map from a compact space to a Hausdorff space, thus closed.

It follows that a proper map $f:X\to Y$ to a locally compact Hausdorff space $Y$ is closed. If $f$ is injective, then it’s an embedding. Also note that a subset $B$ of $Y$ intersecting every compact set in a finite set has no accumulation points, i.e. $B$ is closed and discrete. So if $A$ is closed and discrete, then so is $f(A)$, and this can be proven with the argument in your post.

In (1) the answer is negative even for a compact case. Let $X=[0,1]^2$, $Y=[0,1]$, and $f:X\to Y$ be the projection onto first coordinate. Put $$D=\{(1/n,1/n):n\in\Bbb N\}\cup\{(0,1)\}.$$ Then the set $D$ is discrete, but its image $f(D)$ is a convergent sequence $$\{1/n:n\in\Bbb N\}\cup\{0\}.$$

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