Intereting Posts

Show that the set of functions under composition is isomorphic to $S_3$
Prove that the congruence $x^2 \equiv a \mod m$ has a solution if and only if for each prime $p$ dividing $m,$ one of the following conditions holds
If $n$ is an odd natural number, then $8$ divides $n^{2}-1$
How many elements of order $k$ are in $S_n$?
Has this phenomenon been discovered and named?
Find the sum of $\sum 1/(k^2 – a^2)$ when $0<a<1$
Compute integral of a Lebesgue measurable set
$f$ differentiable, $f(x)$ rational if $x$ rational; $f(x)$ irrational if $x$ irrational. Is $f$ a linear function?
Embeddings are precisely proper injective immersions.
Generalized Fourier series in $L^2$ that do not converge pointwise a.e.
Importance of determining whether a number is squarefree, using geometry
Frequency of Math Symbols
When is it possible to have $f(x+y)=f(x)+f(y)+g(xy)$?
History Behind Integral Error Between $\pi$ and $22/7$
Line integral over ellipse in first quadrant

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):

- Example of a Borel set that is neither $F_\sigma$ nor $G_\delta$
- Choosing a text for a First Course in Topology
- Proof of the Banach–Alaoglu theorem
- No Smooth Onto Map from Circle to Torus
- Uniqueness for a covering map lift: is locally connected necessary?
- Every metrizable space with a countable dense subset has a countable basis

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.

- Are all Infinite Simplicial Complexes non-compact?
- Real Analysis: open and closed sets
- Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?
- Locally Compact Hausdorff Space That is Not Normal
- Motivation for the importance of topology
- Weak Hausdorff space not KC
- 1-1 correspondence between and
- A set which is neither meagre nor comeagre in any interval.
- A “non-trivial” example of a Cauchy sequence that does not converge?
- Does local convexity imply global convexity?

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.

- How do you prove that vectors are linearly independent in $ \mathcal{C}$?
- Let $p$ be a prime, then does $p^{\alpha} \mid |G| \Longrightarrow p \mid Aut(G)$?
- Twenty questions against a liar
- Every Group is a Fundamental Group
- Can there be generalization of Monty Hall Problem?
- Twin integrals, would like to know them precise.
- Is this sum related to the Gregory's limit?
- Why are harmonic functions called harmonic functions?
- Isomorphisms between Normed Spaces
- What is the result of $\bigcap_{n=1}^{\infty}{(-1/n; 1/n)}$
- Axiom of Choice: What exactly is a choice, and when and why is it needed?
- Is a Fourier transform a change of basis, or is it a linear transformation?
- Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$
- Is composition of covering maps covering map?
- Show that $\int_0^\infty \frac{x\log(1+x^2)}{e^{2\pi x}+1}dx=\frac{19}{24} – \frac{23}{24}\log 2 – \frac12\log A$