Limit Point of Sequence may only be Adherent Point of Range

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

Let $\sequence {x_n}$ be a sequence in $A$.

Let $\alpha$ be a limit of $\sequence {x_n}$.

Then $\alpha$ may be only an adherent point of $A$ and not a limit point of $A$.

Proof

Let $T = \struct {S, \tau}$ be the discrete space on $S$.

Let $x \in S$.

Then by definition of discrete space:

$U = \set x$ is an open set of $T$.

Consider the sequence $\sequence {x_n}$ defined as:

$\forall n \in \N: x_n = x$

That is:

$\sequence {x_n} = \tuple {x, x, x, \ldots}$

From Limit Point of Sequence in Discrete Space not always Limit Point of Open Set:

$x$ is not a limit point of $U$.

But from Limit Point of Sequence is Adherent Point of Range:

$x$ is an adherent point of $\set x$.

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Limit Points