# Tag Info

### Understanding the witness property in the Henkin construction

You're correct that the two versions are equivalent when $T$ is complete (maximal), and your proposed argument (alternating between completing the theory and adding witnesses) works just fine. In fact,...
• 80.1k
Accepted

### Heine-Borel for Finite Dimensional Normed Vector Spaces

Throughout, let $\Bbb{F}$ be either $\Bbb{R}$ or $\Bbb{C}$. Recall that if $X,Y$ are normed vector spaces over $\Bbb{F}$ and $T:X\to Y$ is linear, then the following statements are equivalent: $T$ is ...
• 60.2k
Accepted

### Pointwise convergence of almost periodic function

Put $K=\overline{\{f_t\}_{t\in \mathbb{R}}}^{\|\cdot\|_{\infty}}$. Since the set $K$ is compact in the given set $X$ of functions from $\mathbb R$ to $\mathbb R$, with the topology $\tau_\infty$, ...
• 95.4k

### Is $X\simeq [0,1]$?

I'm not sure why, but this question was answered and then the answer was deleted less than an hour later. I will post this answer again, but if the original answerer undeletes and lets me know, then I'...
• 104k
1 vote

Here is a counterexample if $A$ is disconnected: Let $A_p=\mathbb{R} \times B(p,1)$, and $A = A_{-2} \cup A_2$. Define $f(t,x) = \begin{cases} 0, & (t,x) \in A, t \le 0 \\ 0, & (t,x) \in A_{-2}... • 175k 1 vote ### Why do we need step 4. in this proof of the compactness of$[0, 1]$? In your question you restate Spivak's proof. It seems that you already found the answer to Q1. However, your formulation of$(4)$is misleading: There is an open interval$J$,$\alpha\in J\subseteq U$... • 4,749 1 vote Accepted ### Why do we need step 4. in this proof of the compactness of$[0, 1]$? The purpose of step 4 is to provide an interval which contains$\alpha$and some point$x \in A$. As you say,$U$may be made of disconnected intervals, so we need to first pick an interval within$U$... • 1,253 1 vote Accepted ### Quasicompactness of$\operatorname{Proj}A$I have figured this out. By hypothesis we have that: $$\sqrt{A_+}=\sqrt{\langle f_1,\dots, f_n\rangle}$$ hence: $$V(A_+)=\bigcap_{i=1}^nV(f_i)$$ By taking compliments, it follows that:$$\operatorname{... • 3,994 1 vote Accepted ### If$(X,\tau)$is compact and$\mathcal{F}\subset C(X)$is separating points, then the weak topology generated by$\mathcal{F}$is equal to$\tau$As$\sigma\subset \tau$, you know that the identity map$1_X\colon(X,\tau)\to (X,\sigma)$is continuous. Clearly it is a bijection. You are given that$(X,\tau)$is compact.$(X,\sigma)\$ is Hausdorff,...
• 13.2k

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