Math 440, Fall 2014, Assignment 9

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

Carefully define the following terms, then give one example and one non-example of each:[edit]

Problems 16A(1) and 16B(1).
Show that any open ball in $\mathbb{R}^n$ is homeomorphic to all of $\mathbb{R}^n$. (Hint: first use translation and dilation maps to show that any open ball is homeomorphic to the open unit ball centered at the origin. Then make a homeomorphism from the open unit ball to all of $\mathbb{R}^n$ by multiplying each point by a certain scalar, depending on the point's distance from the origin.)
Show that any open subset of a topological manifold is again a topological manifold in the subspace topology.
Recall that $GL_n(\mathbb{R})$ denotes the set of invertible $n\times n$ matrices with real entries. Show that $GL_n(\mathbb{R})$ is a topological manifold with respect to the subspace topology it inherits from $\mathbb{R}^{\left(n^2\right)}$.

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Normal Space:
A space $X$ is normal iff whenever $A$ and $B$ are disjoint closed sets in $X$, there are disjoint open sets $U$ and $V$ with $A \subset U$ and $B \subset V$.
Example:
Non-Example:
$T_4$-Space:
A $T_4$ space is a normal $T_1$ space.
Example:
Non-Example:
Urysohn Function:
A Urysohn function is a continuous function on $X$ created by an application of Urysohn's lemma.
Example:
Non-Example:
Second Countable Space:
A space $X$ is second countable iff it has a countable base.
Example:
Non-Example:
Separable Space:
A space is separable iff it is has a countable dense subset.
Example:
Non-Example:
Topological Manifold:
A topological manifold is a second countable Hausdorff space $X$ for which every point $x \in X$ has a neighborhood homeomorphic to $\mathbb{R}^n$, for some fixed $n$ (the same $n$ at all points).
Example: $\mathbb{R}^n$ itself.
Non-Examples: The line with two origins is second countable and locally Euclidean, but it is not Hausdorff. The disjoint union of uncountably many copies of $\mathbb{R}$ is Hausdorff and locally Euclidean, but not second countable. The union of the two coordinate axes in $\mathbb{R}^2$ is Hausdorff and second countable, but not locally Euclidean (at the origin).

Urysohn's Lemma
A space is normal iff whenever $A$ and $B$ are disjoint closed sets in $X$, there is a continuous function $f: X \rightarrow [0,1]$ with $f(A) = 0$ and $f(B) = 1$.
Tietze Extension Theorem
$X$ is normal iff whenever $A$ is a closed subset of $X$ and $f:A\rightarrow \mathbb{R}$ is continuous, there is an extension of $f$ to all of $X$.
Theorem Relating Second Countable $T_3$-spaces to $T_4$-spaces.
A second countable $T_3$ space is $T_4$.

A. Prove that every subspace of a first countable space is first countable.
Suppose $X$ is first countable, and $A$ is a subset of $X$. Suppose $a \in A$. Then $a$ has a countable neighborhood base $\mathscr{U}_a$. Define $\mathscr{V}_a = \{U \cap A| U \in \mathscr{U}_a\}$. Suppose $L$ is any neighborhood of $a$ in $A$. Then $L = M \cap A$ for some $M$. Clearly $M$ is a nhood of $a$ in $X$, so there is a $U \in \mathscr{U}_a$ such that $U \subset M$. Then $V = U \cap A \subset M \cap A = L$, so every nhood of $a$ contains an element of $\mathscr{V}_x$, so $\mathscr{V}_a$ is a nhood base of $a$.
B.
The ball is centered at $\vec{x}$. Translate it by $-\vec{x}$. It is of radius $r$. Scale it by $1/r$. Then scale each point by $1/1-|\vec{v}|$.
An open subset of a topological manifold is homeomorphic to an open subset of $\mathbb{R}^n$. Any point in this subspace then has a neighborhood homeomorphic to an $n$-ball. Thus, by the previous problem, it has a neighborhood homeomorphic to $\mathbb{R}^n$.