Math 311 Definitions and Theorems Flashcards

Critical definitions and theorems from Math 311 Lectures 7 through 10, covering real analysis topics such as topology, compactness, connectedness, and continuity.

How is an $\epsilon$-ball centered at $x_0$ defined in $\mathbb{R}$?

B_\epsilon(x_0) = \{x \in \mathbb{R} \mid d(x, x_0) < \epsilon\}

What is the definition of an interior point of a set $A \subset \mathbb{R}$?

A point $x \in A$ is an interior point if $\exists \epsilon > 0$ such that $B_\epsilon(x) \subset A$.

How is the interior of a set $A$, denoted $A^\circ$, defined?

A^\circ = \{x \in A \mid \exists \epsilon > 0 \text{ such that } B_\epsilon(x) \subset A\}

When is a set $A$ considered open?

A set $A$ is open if $A = A^\circ$, meaning all its points are interior points.

What is the definition of a limit point for a set $A \subset \mathbb{R}$?

A point $x \in A$ is a limit point if $\exists (x_n) \subset A$ such that $x_n \rightarrow x$ and $x_n \neq x$ for all $n$.

How is an isolated point defined?

A point is called an isolated point if it is not a limit point of the set $A$.

How is the closure ($\overline{A}$) of a set $A$ defined?

$$\overline{A} = \{x \in \mathbb{R} \mid x \text{ is a limit point of } A\} \cup \{\text{isolated points}\}$$

What is the condition for a set $A$ to be closed?

A set $A$ is closed if $A = \overline{A}$.

What constitutes an open cover of a set $A$?

A family of open sets $O_\lambda, \lambda \in \Gamma$ such that $A \subset \bigcup_{\lambda \in \Gamma} O_\lambda$.

What is the definition of a compact set $A \subset \mathbb{R}$?

A set is compact if any open cover of the set has a finite subcover.

According to the Heine-Borel Theorem, what three conditions are equivalent for a set $K \subset \mathbb{R}$?

1. $K$ is compact. 2. $K$ is sequentially compact (every sequence has a convergent subsequence). 3. $K$ is closed and bounded.

How is the diameter of a set $A$ defined?

$$\text{diam } A = \sup_{x,y \in A} \{|x - y|\}$$

What is the Nested Compact Sets Theorem?

If $K_n$ is nonempty and compact for all $n$, and $K_1 \supset K_2 \supset K_3 \supset ...$, then $\bigcap_{n=1}^\infty K_n$ is nonempty.

What is the definition of a disconnected set $X$?

$X$ is disconnected if $\exists A, B$ open in $X$ such that $A \cap B = \emptyset$, $A \cup B = X$, and $A, B \neq \emptyset$.

How is a set $A$ defined as 'open in $X$' within the subset topology?

$A$ is open in $X$ if $\exists U \subset \mathbb{R}$ open in $\mathbb{R}$ such that $A = U \cap X$.

What is a 'clopen' set?

A set $A \subset X$ is clopen if it is both open and closed.

What is the $\epsilon-\delta$ definition of $\lim_{x \rightarrow c} f(x) = L$?

$\forall \epsilon > 0, \exists \delta > 0$ such that $|x - c| < \delta \implies |f(x) - L| < \epsilon$.

What is the Divergence Criterion for limits of functions?

If $x_n \rightarrow c$ and $y_n \rightarrow c$, but $f(x_n)$ and $f(y_n)$ have different limits, then $\lim_{x \rightarrow c} f(x)$ does not exist.

What is the definition of a function $f$ being continuous at a point $c \in A$?

$f$ is continuous at $c$ if $\lim_{x \rightarrow c} f(x) = f(c)$, which requires the limit to exist.

What is the topological definition of continuity for a function $f: A \rightarrow B$?

$f$ is continuous if and only if for all open sets $U \in B$, the inverse image $f^{-1}(U)$ is open in $A$.