(1)A is a point set, and x0 is a fixed point (which may not belong to A). If any neighborhood of x0 contains an infinite number of points in A, it is said that x0 is a gathering point of A. ..
(2)A is a point set, and x0 is a fixed point (not necessarily belonging to A). If any neighborhood of x0 contains a point different from X in A, it is said that x0 is the aggregation point of A. ..
Obviously (1) deduces (2). If we take any neighborhood, there are infinitely many points in this neighborhood, and of course there are also points different from x0.
(2) Push (1) can be proved by construction method:
If δ 1 = 1, there is x1∈ uhhapy (x0, δ 1)∩a,
If δ2 = min( 1/2, ㄧx0-x 1ㄧ), there is x2∈U happy (x0, δ2)∩A, x2≠x 1.
......
If δn = min( 1/n, ㄧx0-xn- 1ㄧ), then xn∈U happy (x0, δn)∩a exists, and xn and x 1, x2 exist. ...
In this way, you can get infinite points.
In the proof, some skills will be used when constructing infinite points. You can draw a number axis and make a chart to help you understand.
When studying functional analysis and point set topology in the future, you will also encounter concepts such as gathering point, open set and closed set. In functional analysis, there are three similar equivalent definitions of aggregation points in metric space. In the point set topology, there is only one definition similar to (2), which can only be defined by points.