## Reference for compact embedding between (weighted) Holder space on $\mathbb{R}^n$

Suppose $$0<\alpha<\beta<1$$, and $$\Omega$$ is a bounded subset of $$\mathbb{R}^n$$. Then the Holder space $$C^{\beta}(\Omega)$$ is compactly embedded into $$C^{\alpha}(\Omega)$$. But if $$\Omega=\mathbb{R}^n$$, then the compact embedding is not true.

However, if we consider the weaker weighted Holder space $$C^{\alpha, -\delta}(\mathbb{R}^n)$$ (for any $$\delta>0$$) instead of $$C^{\alpha}(\mathbb{R}^n)$$. Then is $$C^{\beta}(\mathbb{R}^n)$$ compactly embedded to $$C^{\alpha, -\delta}(\mathbb{R}^n)$$?

Here $$\|f\|_{C^{\alpha, -\delta}}=\|(1+|\cdot|^2)^{-\frac{\delta}{2}}f\|_{C^{\alpha}}.$$

I could not find a precise reference from some books on functional analysis. Any comment is welcome.

## $U=\{x\in \mathbb{R^n} :g_i(x)\leq 0 : \text{for} \quad i=1,…,m \}$ is closed

Let $$U=\{x\in \mathbb{R^n} :g_i(x)\leq 0 : \text{for} \quad i=1,…,m \}$$ be a subset of $$\mathbb{R^n}$$.

with $$(g_i)_{1\leq i \leq n}$$ are convex functions from $$\mathbb{R^n}$$ in $$\mathbb{R}$$.

we want to prove that $$U$$ is a closed set.

## Connected and homogeneous $T_2$-space not homeomorphic to a subset of $\mathbb{R}^n$

What is an example of an connected and homogeneous $$T_2$$-space $$(X,\tau)$$ with $$|X|=2^{\aleph_0}$$ such that for no $$n\in\mathbb{R}$$ the space $$(X,\tau)$$ is homeomorphic to a subspace of $$\mathbb{R}^n$$?