Real Analysis: Topology of the Real Line

Find a finite subcover for the given open cover of an interval, if possible.

Prove that a given set is not compact by finding an open cover with no finite subcover.

Prove the classic theorem that says a subset $K$ of $\mathbb{R}$ is compact if and only if it is closed and bounded.

Demonstrate that a set in $\mathbb{R}$ is disconnected if open sets $U_1$ and $U_2$ can be found such that the intersection of $e$ with both $U_1$ and $U_2$ are non-empty, and their union covers all of $e$, while the intersection of these parts is empty.

Determine if a punctured interval is a disconnected set by identifying appropriate open sets.

Illustrate the disconnection of two sets in bb{R}^2 using open sets to separate them.

Determine if the set $(-2, -1) \cup (-1, 0)$ is connected or disconnected.

Define when a set is considered connected in terms of open sets.

Prove that if a subset of $\R^P$ is compact, then it is closed and bounded.

Prove that a closed interval [c, d] of real numbers is a compact set.

Show that if a set $A$ is bounded and closed, then it is compact, according to the Heine-Borel theorem.

Find the limit points and derive the set of a set $A$ in the topology $\tau$ on a set $X$, where $X = \{ A, B, C, D, E \}$.

Prove that a subset $A$ of a topological space $X$ is closed if and only if it contains all of its limit points.

Determine if the set of integers is open, closed, or compact.

Determine if the set is open, closed, or compact: the union of the reciprocals of natural numbers with the set containing zero.

Determine if the set of real numbers is open, closed, or compact.

Determine if the set is open, closed, or compact: the union of an open interval (0, 1) and a closed interval [3, 4].

Determine if the set of rational numbers is open, closed, or compact.

Determine if a singleton set containing only the number seven is open, closed, or compact.

Determine if the interval $[1, 3)$ is an open set in the metric space of real numbers defined as $x = \{1 \leq x < 3\} \cup \{x > 4\}$, where the metric is the standard distance function.