$X$ is a topology space and $A=\{x_1 , x_2 , … , x_n , … \}$ is a subset of $X$ , $y$ is a limit point of $A$ .
Can we show that for each $N$ , $y$ is also a limit point of $A'=\{x_N ,x_{N+1}, … \}$ ?
It is obvious that the conclusion hold when $X$ is a metric space . But it might not be valid for all topology space since if it is true , then we can use it to prove : If every infinite subset in $X$ has a limit point , then $X$ is compact . (Which is not true for all topology space )
Suppose there is family of open sets $O_1,\ldots,O_n,\ldots$ such that $X \subset \cup O_n$ but there is no finite subcovering. Construct the sequence $A=(x_n)$ such that $x_n \in X \setminus \bigcup_{i=1}^n O_i $ . But our assumption $A$ has an limit point $x$ , we say that $x \in O_N$ for some $N$ . Let $$A'=\{x_N ,x_{N+1}, … \}$$ then we can see $x$ is not a limit point of $A'$ , since $x \in O_N $ but $O_N$ contains no element of $A'$ .
My question:
Is there any counterexample such that : $X$ is a topology space , every infinite subset of $X$ has a limit point but $X$ is not compact .
Best Answer
That $x$ is a limit point of $A$ implies that it is a limit point of $A\setminus F$, where $F \subseteq A$ is finite, will hold in any $T_1$ space (so when singletons are closed). That answers the first part of your question.
In a $T_1$ space, being a limit point of $A$ (every neighbourhood of $x$ intersects $A\setminus\{x\}$) is equivalent to $x$ being an $\omega$-limit point of $A$ (every neighbourhood of $x$ contains infinitely many points of $A$).
"Every infinite set $A$ has an $\omega$-limit point" is in general spaces equivalent to "every countable open cover of $X$ has a finite subcover", i.e. countable compactness, which for metrisable spaces indeed is equivalent to compactness but in general not: consider $\omega_1$, the first uncountable ordinal in the order topology, or $\{0,1\}^\mathbb{R}\setminus \{\underline{0}\}$, which are countably compact but not compact. The first is also sequentially compact, first countable, hereditarily normal etc., so even for very nice spaces these notions need not be equivalent.