# Are singletons limit points?

So I just started learning topology and I came across this problem.

Consider the topological space $$(X,\tau)$$where the set $$X=\{a,b,c,d,e\}$$ and $$\tau=\{X,\emptyset,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e\}\}$$

Determine the limit points of

i) $$\{a\}$$

Is $$a$$ a limit point for itself since the singleton a is an open set in the topology. From the definition of limit point of a set every neighbourhood of the point must contain a point of the set that id different from a however when considering the singleton there are no different points of a so what is the correct answer im very confused.

• It is not a limit point. Jan 22 at 9:24

The limit points of a set $$S$$ are not an intrinsic property of $$S$$. They depend on how the set $$S$$ is immersed into an ambient topological space $$X$$.

To answer your question, you have to check if there are points $$x\in X$$ such that every punctured neighborhood of $$x$$ (i.e., the result of removing from an open neighborhood of $$x$$ the point $$x$$) passes through $$a$$.

• Now the punctured neighborhoods of $${a}$$ cannot contain $$a$$.
• So let's examine the punctured neighborhoods of $${b}$$. These are $$\{c,d,e\}$$ and $$X-\{b\}$$. At least one does not contain $$a$$. So $$b$$ is not a limit point of $$\{a\}$$.
• The punctured neighborhoods of $${c}$$ are $$\{d\}$$ and others. But you already found one that does not contain $$a$$ so $$c$$ cannot be a limit point of $$a$$.

And so on...

If you want some intuition about the concept of a limit point, then you can think that an element $$x\in X$$ is a limit point of a subset $$S\subseteq X$$ if $$x$$ can be well approximated by $$S-\{x\}$$.

• just being curious isn't $\{a,c,d,b\}$ also a nbhd of b? Jan 22 at 11:26
• No, the neghbs of b are all superset of {b,c,d,e}. Because they must contain an open set containing b.
– Kosh
Jan 22 at 11:26
• isn't $\{a,c,d\}$ open in this topology? Jan 22 at 11:30
• Yes, it is open. You listed the open sets. But this open set does not contain b.
– Kosh
Jan 22 at 12:04

The set of limit points of $$\{a\}$$ is empty.