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:

1. Normal space.
2. $T_4$-space.
3. Urysohn function (for a pair of subsets $A,B\subseteq X$).
4. Second countable space.
5. Separable space.
6. Topological manifold.

Carefully state the following theorems (you need not prove them):

1. Urysohn's Lemma.
2. Tietze Extension Theorem.
3. Theorem relating second countable $T_3$-spaces to $T_4$-spaces.

Solve the following problems:

1. Problems 16A(1) and 16B(1).
2. 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.)
3. Show that any open subset of a topological manifold is again a topological manifold in the subspace topology.
4. 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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