Last updates: 7 December 2009
Define the following and give an example for each:
 
Define the following and give an example for each:
 
Define the following and give an example for each:
 
Prove that
$\mathcal{B}$ and
is a basis of
$\mathcal{T}$ if and only if
$\mathcal{B}$ satisfies:
if
$x\in X$
then
$\mathcal{B}\left(x\right)=\{B\in \mathcal{B}\hspace{0.17em}\hspace{0.17em}x\in B\}$
is a fundamental system
of neighborhoods of
$x$.
 
Let
$X$ and
$Y$ be metric spaces. Define the topology on
$X$ and
$Y$.
Prove that
$f:X\to Y$
is continuous as a function between metric spaces
if and only if
$f:X\to Y$
is continuous as a function between topological spaces.
 
Define the following and give an example for each:
 
Define the following and give an example for each:
 
Let
$X$ be a Hausdorff topological space and let
$K$ be a compact subset of
$X$. Show that
$K$ is closed.
 
Let
$X$ be a metric space. Show that
$X$ is Hausdorff and has a countable basis.
 
Let
$X$ be a metric space and let
$K$ be a compact subset of
$X$. Show that
$K$ is closed and bounded.
 
Let
$X$ be a metric space and let
$E$ be a subset of
$X$. Show that
$E$ is compact if and only if every infinite subset of
$E$ has a limit point in
$E$. (What is the definition of limit point???)
 
Let
$K$ be a subset of
${\mathbb{R}}^{n}$. Show that
$K$ is compact if and only if
$K$ is closed and bounded.
 
Let
$X$ and
$Y$ be topological spaces and let
$f:X\to Y$ be a continuous function. Show that
if
$X$ is connected then
$f\left(X\right)$ is connected.
 
Let
$E\subseteq \mathbb{R}$. Show that
$E$ is connected if and only if the
set $E$ satisfies
if
$x,y\in E$ and
$z\in \mathbb{R}$
and
$x<z<y$
then
$z\in E$.
 
(Intermediate Value Theorem) Let
$f:[a,b]\to \mathbb{R}$ be a continuous function. Show that
if
$z\in \mathbb{R}$
and
$f\left(a\right)<z<f\left(b\right)$
then there exists
$c\in (a,b)$
such that
$f\left(c\right)=z$.
 
Let
$X$ and
$Y$ be topological spaces and let
$f:X\to Y$ be a continuous function. Show that
if
$X$ is compact then
$f\left(X\right)$ is compact.
 
Let
$D$ be a closed bounded subset of
$\mathbb{R}$ and let
$f:D\to \mathbb{R}$ be a continuous function.

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561575; arXiv:math/9909077v2, MR1828302 (2002e:20083)