Intereting Posts

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PDF of a sum of exponential random variables
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What are the main uses of Convex Functions?
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Statistics: Bertrand's Box Paradox
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Approximating Borel sets by finite unions of intervals
Simplification of expressions containing radicals
Distribution of sums
Showing that $n$ exponential functions are linearly independent.
Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.
How to solve this nonlinear constrained optimization problem

My proof:

Let $X$ be an space with the order topology, $x \in X$ and $F$ a closed set that does not contain $x$. Then, the set $X-F$ is an open set that contains $x$, hence there is an open set (basic) $(a,b)$ such that $x \in (a,b)\subseteq X-F$. Then $(a,b)$ and $(-\infty,a) \cup (b,\infty)$ are open disjoint sets that separate $x$ and $F$.

I am not sure because I’ve seen other proofs and they are much more complicated, like this one (source):

- Two continuous functions that are the same in the rationals.
- Is Every (Non-Trivial) Path Connected Space Uncountable?
- Separation axioms in uniform spaces
- Proof that a perfect set is uncountable
- Existence of a continuous function which does not achieve a maximum.
- Separating points from open sets in a compact space without isolated points

Besides, in that proof I don’t understand when do they use the fact that a point is closed in a Hausdorff space. I also found that the last union is not disjoint, I double check that, but I may be missing something.

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- Continuous Deformation Of Punctured Torus
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- Every neighborhood of identity in a topological group contains the product of a symmetric neighborhood of identity.

Your proof wouldn’t work as while your (a,b) could exist, there is nothing to say that a and b aren’t in F. So your open set wouldn’t necessarily contain F.

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