Intereting Posts

Is $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$ a differentiable submanifold?
Show $A_n$ has no subgroups of index 2
What is the precise definition of $i$?
A proof of a theorem on the different in algebraic number fields
A collection of sequences that cannot all be made to converge
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
Two exponential terms equation solution
Capelli Lemma for polynomials
About the interior of the union of two sets
How to show every subgroup of a cyclic group is cyclic?
Proof of the product rule. Trick. Add and subtract the same term.
Automorphism group of the Alternating Group – a proof
Constructing a choice function in a complete & separable metric space
Interchange supremum and expectation
Finding minimum $\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}$

Possible Duplicate:

How would one go about proving that the rationals are not the countable intersection of open sets?

As the topic, prove that set $S$ of rational numbers in the interval (0,1), cannot be expressed as the intersection of a countable collection of open sets.

- When is Stone-Čech compactification the same as one-point compactification?
- Pullback of a covering map
- Non-isomorphic Group Structures on a Topological Group
- A weak version of Markov-Kakutani fixed point theorem
- How do I show that $f: [0,1) \to S^1$, $f(t) = (\cos(2\pi t), \sin(2\pi t))$ is not a homeomorphism?
- if every continuous function attains its maximum then the (metric) space is compact

- Example of different topologies with same convergent sequences
- Definition of a manifold
- The integral of a closed form along a closed curve is proportional to its winding number
- Basis for a topology with a countable number of sets
- Exercise on compact $G_\delta$ sets
- Can $k$ be dense in $k$? where $p_xq_y-p_yq_x \in k^*$.
- Compact sets are closed?
- identify the topological type obtained by gluing sides of the hexagon
- Is every countable dense subset of $\mathbb R$ ambiently homeomorphic to $\mathbb Q$
- Cofinite\discrete subspace of a T1 space?

This follows immediately from the Baire category theorem. Suppose that $\Bbb Q\cap(0,1)=\bigcap_{n\in\Bbb N}U_n$, where each $U_n$ is open. For each $q\in\Bbb Q\cap(0,1)$ let $V_q=(0,1)\setminus\{q\}$. Then $$\{U_n:n\in\Bbb N\}\cup\{V_q:q\in\Bbb Q\cap(0,1)\}$$ is a countable family of dense open subsets of $(0,1)$ whose intersection is empty. Since $(0,1)$ is locally compact, however, the Baire category theorem ensures that the intersection of a countable family of dense open sets is dense in $(0,1)$.

- FLOSS tool to visualize 2- and 3-space matrix transformations
- “Constrained” numerical solutions of ODEs with conservation laws?
- What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.
- How to prove the Cone is contractible?
- A WolframAlpha error?
- Action of a matrix on the exterior algebra
- Homotopy groups of $S^2$
- How to compute the $n_{th}$ derivative of a composition: ${\left( {f \circ g} \right)^{(n)}}=?$
- Criteria for swapping integration and summation order
- Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$
- Show that a process is no semimartingale
- “Nice” well-orderings of the reals
- Find laurent expansion of $\frac{z-1}{(z-2)(z-3)}$ in annulus {$z:2<|z|<3$}.
- Is $2^\alpha=2^\beta\Rightarrow \alpha=\beta$ a $\sf ZFC$-independence result?
- How to draw ellipse and circle tangent to each other?