Some questions/answers I find particularly attractive:

- $f\colon\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(f(x))=x^2$ for all $x$?
- Where do Cantor sets naturally occur?
- Connected metric spaces with disjoint open balls
- locally self-similar topologies
- Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?
- Is this Fourier like transform equal to the Riemann zeta function?
- Connectedness of the boundary
- Universally measurable sets of $\mathbb{R}^2$
- Lifting local compactness in covering spaces
- What are the (non-piecewise) linear manifolds?
- Highest DeRahm Cohomology
- Shrinking Group Actions
- What is the Sequence that Maximizes this Distance?
- systole of space projective 3-dimensional
- fundamental group of $U(n)$
- Geodesic on Spheres
- Identification of integration on smooth chains with ordinary integration
- Is there a compact, connected manifold $M$ with boundary $S^2$ not diffeomorphic to $D^3$?
- What is $T\mathbb{S}^2$?
- Functions space of discrete space: how does taking quotients lead to noncommutativity?
- Half the rationals?
- Prove that a compact metric space can be covered by open balls that don't overlap too much.
- Sheafs and closed immersion
- Can an infinite cardinal number be a sum of two smaller cardinal number?
- cover a sphere with 3 open semispheres
- Prove the following inequality
- If an unary language exists in NPC then P=NP
- Continuous injective map $f:\mathbb{R}^3 \to \mathbb{R}$?
- Intuition for cofibration
- $\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.
- pseudo-inverse to the operation of turning a metric space into a topological space