Introduction To Topology Mendelson Solutions -

Let $A \subseteq X$. Suppose that $A$ is open. Then, for each $a \in A$, there exists $r_a > 0$ such that $B(a, r_a) \subseteq A$. This implies that $A = \bigcup_{a \in A} B(a, r_a)$.

"Introduction to Topology" by Bert Mendelson is a classic textbook that provides a rigorous and concise introduction to the field of topology. The book was first published in 1963 and has since become a standard reference for students and researchers. The book covers the basic concepts of point-set topology, including topological spaces, continuous functions, compactness, and connectedness. Introduction To Topology Mendelson Solutions

Let $A \subseteq X$. We need to show that $\overline{A}$ is the smallest closed set containing $A$. First, we show that $\overline{A}$ is closed. Let $x \in X \setminus \overline{A}$. Then, there exists an open neighborhood $U$ of $x$ such that $U \cap A = \emptyset$. This implies that $U \subseteq X \setminus \overline{A}$, and hence $X \setminus \overline{A}$ is open. Therefore, $\overline{A}$ is closed. Let $A \subseteq X$

Conversely, suppose that $A = \bigcup_{a \in A} B(a, r_a)$ for some $r_a > 0$. Let $x \in A$. Then, there exists $a \in A$ such that $x \in B(a, r_a)$. This implies that there exists an open ball around $x$ that is contained in $A$, and hence $A$ is open. This implies that $A = \bigcup_{a \in A} B(a, r_a)$