Why does $ U \cap (X − \{x_1, . . . , x_m\})$ not intersect the set $A-\{x\}$ at all?

Question

Munkre Topology section 17

Theorem 17.9.

Let X be a space satisfying the $T_1$ axiom; let $A$ be a subset of $X$. Then the point $x$ is a limit point of $A$ if and only if every neighborhood of $x$ contains infinitely many points of A.

My question is why does $ U \cap (X − \{x_1, \dots , x_m\})$ not intersect the set $A-\{x\}$ at all?

Answer 1

$(A-\{x\})\cap U=\{x_1,\dots,x_m\}$

so:

$(A-\{x\})\cap U\cap(X-\{x_1,\dots,x_m\})=\{x_1,\dots,x_m\}\cap(X-\{x_1,\dots,x_m\})=\varnothing$