Show that the unit sphere bundle of the $r$-fold direct sum of the tautological (universal) complex line bundle over the space $\mathbb{C}P^{\infty}$ is homotopically equivalent to $\mathbb{C}P^{r-1}$.

Let $A$ be a closed subset of $\mathbb{R}^{n}$ and let $B$ be the set of all those points $b \in \mathbb{R}^{n}$ for which there exists exactly one point $a_{0}\in A $ such that $|a_{0}-b|= \inf_{a\in A}|a-b|$. Prove that $B$ is dense in $\mathbb{R}^{n}$; that is, the closure of $B$ is $\mathbb{R}^{n}$

Does a circle have a subset which is topologically closed and which contains exactly one point of each pair of diametrically opposite points?

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

Show that if a connected, nowhere zero sectional curvature of Riemannian manifold, where symmetric (1,1)-tensor of the Levi-Civita connection covariant derivative vanishes, then the tensor is constant times the unit tensor. (translated by j___d)

Compute $\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}$

Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.

A cube is made by soldering twelve $ 3$-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is: $ \textbf{(A)}\ 24\text{ in.}\qquad \textbf{(B)}\ 12\text{ in.}\qquad \textbf{(C)}\ 30\text{ in.}\qquad \textbf{(D)}\ 18\text{ in.}\qquad \textbf{(E)}\ 36\text{ in.}$

Prove that a Hausdorff space X is countably compact iff for every open cover $\cal {U}$ there is a finite set $A \subset X$ such that $ \bigcup \{U \in {\cal U} : U \cap A \neq \emptyset \} = X$.

Suppose that the closed subset $K$ of the sphere $$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$ is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between $$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$ and $K$ is less than $\varepsilon$.

Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$? (We use the metric $$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$ to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$.)

Let $ M^n \subset \mathbb{R}^{n\plus{}1}$ be a complete, connected hypersurface embedded into the Euclidean space. Show that $ M^n$ as a Riemannian manifold decomposes to a nontrivial global metric direct product if and only if it is a real cylinder, that is, $ M^n$ can be decomposed to a direct product of the form $ M^n\equal{}M^k \times \mathbb{R}^{n\minus{}k} \;(k<n)$ as well, where $ M^k$ is a hypersurface in some $ (k\plus{}1)$-dimensional subspace $ E^{k\plus{}1} \subset \mathbb{R}^{n\plus{}1} , \mathbb{R}^{n\minus{}k}$ is the orthogonal complement of $ E^{k\plus{}1}$. [i]Z. Szabo[/i]

X is a compact T2 space such that every subspace of cardinality $\aleph_1$ is first countable. Prove that X is first countable.

Let $G_0, G_1,\ldots$ be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets $V_0,V_1,\ldots$ and some indices $n_0<n_1<\cdots$ such that $V_i\subseteq G_{n_i}$ for every $i\geqslant 0.$

Upper content of a subset $E$ of the plane $\mathbb{R}^{2}$ is defined as $$\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}$$ where $\inf$ is taken over all finite families of sets $E_{1},\dots,E_{n}$ $n\in \mathbb{N}$, in $\mathbb{R}^{2}$ such that $E\subset \bigcup_{i=1}^{n}E_{i}$. Lower content of $E$ is defined as $$\mathcal{K}(E)=\sup\{\text{length}(L) |\, L \text{ is a closed line segment onto which $E$ can be contracted}\}$$. Prove that i) $\mathcal{C}(L)=\text{length}(L)$ if $L$ is a closed line segment; ii) $\mathcal{C}(E) \geq \mathcal{K}(E)$; iii) the equality in ii) is not always true even if $E$ is compact.

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

The Lindelöf number $L(X)$ of a topological space $X$ is the least infinite cardinal $\lambda$ with the property that every open covering of $X$ has a subcovering of cardinality at most $\lambda$. Prove that if evert non-countably infinite subset of a first countable space $X$ has a point of condensation, then $L(X)=\sup L(A)$, where $A$ runs over the separable closed subspaces of $X$. (A point of condensation of a subset $H\subseteq X$ is a point $x\in X$ such that any neighbourhood of $x$ intersects $H$ in a non-countably infinite set.)

Prove that for a $2$-dimensional Riemannian manifold there is a metric linear connection with zero curvature if and only if the Gaussian curvature of the Riemannian manifold can be written as the divergence of a vector field.

Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that (1) All elements of $A_n$ are rational. (2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.

Let $ H$ be an arbitrary subgroup of the diffeomorphism group $ \mathsf{Diff}^\infty(M)$ of a differentiable manifold $ M$. We say that an $ \mathcal C^\infty$-vector field $ X$ is [i]weakly tangent[/i] to the group $ H$, if there exists a positive integer $ k$ and a $ \mathcal C^\infty$-differentiable map $ \varphi \mathrel{: } \mathord{]} \minus{} \varepsilon,\varepsilon\mathord{[}^k\times M\to M$ such that (i) for fixed $ t_1,\dots,t_k$ the map \[ \varphi_{t_1,\dots,t_k} : x\in M\mapsto \varphi(t_1,\dots,t_k,x)\] is a diffeomorphism of $ M$, and $ \varphi_{t_1,\dots,t_k}\in H$; (ii) $ \varphi_{t_1,\dots,t_k}\in H \equal{} \mathsf{Id}$ whenever $ t_j \equal{} 0$ for some $ 1\leq j\leq k$; (iii) for any $ \mathcal C^\infty$-function $ f: M\to \mathbb R$ \[ X f \equal{} \left.\frac {\partial^k(f\circ\varphi_{t_1,\dots,t_k})}{\partial t_1\dots\partial t_k}\right|_{(t_1,\dots,t_k) \equal{} (0,\dots,0)}.\] Prove, that the commutators of $ \mathcal C^\infty$-vector fields that are weakly tangent to $ H\subset \textsf{Diff}^\infty(M)$ are also weakly tangent to $ H$.

For each $M$ m-dimensional closed $C^{\infty}$ set , assign a $G(m)$ in some euclidean space $\mathbb{R}^{q}$. Denote by $\mathbb{R} \mathbb{P}^{q}$ a $q$-dimensional real projecive space. A$G(M) \subseteq \times \mathbb{R} \mathbb{P}^{q}$. The set consists of $(x,e)$ pairs for which $x \in M \subseteq \mathbb {P}^{q} $ and $e \subseteq \mathbb {R}^{q+1}= \mathbb{R}^{q} \times \mathbb{R}$ and $\mathrm{a} (0, \ldots,0,1) \in \mathbb{R}^{q+1}$ in a stretched $(m+1)$-dimensional linear subspace. Prove that if $N$ is a $n$-dimensional closed set $C^{\infty}$, then $P=G(M \times M)$ and $Q=G(M) \times G(N)$ are cobordant , that is, there exists a $(2m+2n+1)$-dimensional compact , flanged set $C^{\infty}$ with a disjoint union of $P$ and $Q$.

Suppose that all subspaces of cardinality at most $ \aleph_1$ of a topological space are second-countable. Prove that the whole space is second-countable. [i]A. Hajnal, I. Juhasz[/i]

Suppose that $X$ and $Y$ are metric spaces and $f:X \longrightarrow Y$ is a continious function. Also $f_1: X\times \mathbb R \longrightarrow Y\times \mathbb R$ with equation $f_1(x,t)=(f(x),t)$ for all $x\in X$ and $t\in \mathbb R$ is a closed function. Prove that for every compact set $K\subseteq Y$, its pre-image $f^{pre}(K)$ is a compact set in $X$.

A ring is the area between two circles with the same center, and width of a ring is the difference between the radii of two circles. [img]http://i18.tinypic.com/6cdmvi8.png[/img] a) Can we put uncountable disjoint rings of width 1(not necessarily same) in the space such that each two of them can not be separated. [img]http://i19.tinypic.com/4qgx30j.png[/img] b) What's the answer if 1 is replaced with 0?

Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.