Intersection of Family/Examples/Size of y-1 lt n and Size of y+1 gt 1 over n

Example of Intersection of Family

Let $I$ be the indexing set $I = \set {1, 2, 3, \ldots}$

Let $\family {T_n}$ be the indexed family of subsets of the set of real numbers $\R$, defined as:

$T_n = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}$

Then:

$\ds \bigcap_{n \mathop \in I} T_n = \openint 0 2$

Proof

From Size of y-1 lt n and Size of y+1 gt 1 over n:

$T_n = \openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n}$

We have that:

$\paren {\openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n} } \subseteq \paren {\openint {1 - \paren {n + 1} } {-1 - \dfrac 1 {n + 1} } \cup \openint {-1 + \dfrac 1 {n + 1} } {1 + \paren {n + 1} } }$

That is:

$T_n \subseteq T_{n + 1}$

and so:

$\ds \bigcap_{n \mathop \in I} T_n = T_1$

Hence:

\(\ds \bigcap_{n \mathop \in I} T_n\)

\(=\)

\(\ds T_1\)

\(\ds \)

\(=\)

\(\ds \openint {1 - 1} {-1 - \dfrac 1 1} \cup \openint {-1 + \dfrac 1 1} {1 + 1}\)

\(\ds \)

\(=\)

\(\ds \openint 0 {-2} \cup \openint 0 2\)

\(\ds \)

\(=\)

\(\ds \O \cup \openint 0 2\)

\(\ds \)

\(=\)

\(\ds \openint 0 2\)

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text {A}$