PMATH 351 Midterm Definitions
$\newcommand{\powerset}[1]{\mathcal P \left({#1}\right)}$
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- Zorns Lemma
- Let $(X, \preceq) \neq \emptyset$ be a poset. Assume that every chain $C \subset X$ has an upper bound. Then $(X, \preceq)$ has a maximal element.
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- Axiom of Choice
- If $I \neq \emptyset$ and $X_\alpha \neq \emptyset$ for $\alpha \in I$, then $\prod_{\alpha \in I} X_\alpha \neq \emptyset$.
- If $X \neq \emptyset$, then there exists a choice function $f: \powerset{X} \setminus \emptyset \to X$ such that $f(A) \in A$ for all $A \in \powerset{X}$.
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- Cantor-Schroeder-Bernstein Theorem
- Assume that $A_2 \subset A_1 \subset A_0$. If $A_2 \sim A_0$, then $A_1 \sim A_0$.
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A set $A \subseteq (X, d)$ is closed $\iff$ whenever ${x_n} \subseteq A$ is such that $x_n \to x_0$ we have $x_0 \in A$.
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A function $f : (X, d_x) \to (Y, d_y)$ is continuous $\iff$ whenever $x_n \to x_0$ in $X$, $f(x_n) \to f(x_0)$ in $Y$.
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The uniform limit of a sequence of continuous functions is continuous.
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Completeness of $C_b(X)$
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- Generalized Weierstrass M-Test
- A normed linear space $(X, |\cdot|)$ is complete $\iff$ whenever $\sum_{n=1}^\infty |x_n|$ converges, so does $\sum_{n=1}^\infty x_n$.
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A subset $A$ of a complete metric space $(X, d)$ is complete in the induced metric $\iff$ it is closed.