Math 440, Fall 2014, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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

1. $T_0$-space.
2. $T_1$-space.
3. $T_2$-space (this is synonymous with Hausdorff space).
4. Regular space.
5. $T_3$ space.
6. Dense subset.

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

1. Theorem characterizing $T_1$-spaces (Theorem 13.4 in the text).
2. Theorem characterizing $T_2$-spaces (Theorem 13.7 in the text).
3. Theorem relating the separation axioms to subspaces, products, and quotients (results of Exercise 13B(1-3) and Theorem 13.8).
4. Theorem concerning the agreement set of two continuous functions (Theorem 13.13).
5. Theorem concerning agreement on a dense set (Corollary 13.14).

Solve the following problems:

1. Problem 13C (this is actually a useful construction in analysis; for example, it is an essential step in the construction of $L^p$ spaces).
2. (The line with two origins) Let $X=\{a,b\}$ be a two-point set with the discrete topology, and give $Y=\mathbb{R}\times X$ the product topology.

(a) Draw a picture of a subspace of $\mathbb{R}^2$ that is homeomorphic to $Y$.

(b) Let $\simeq$ be the equivalence relation on $Y$ whose equivalences classes are the sets $\{(x,a), (x,b)\}$ for $x\neq0$ and the singletons $\{(0,a)\}$ and $\{(0,b)\}$. Where does the quotient space $Y/\simeq$ fall in the separation heirarchy (i.e. is it $T_0, T_1, T_2, T_3,$ or none of these)?

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