Ch. 18 Boundary, Open, Closed 
Word OPEN in mathematics is funny. Here I want to understand it by illustration. Suppose you are in a room and want to go out. If the door is closed, you open it and go through the boundary of the room. In mathematics, however, you can not go through BOUNDARY even if the door is OPEN all the time. You can only get close to it. But never touch it unless specified. BOUNDARY in mathematics is funny, too. It is an invisible door. And CLOSED in mathematics means a closed and locked door though you can touch it.
As to our model disc, the edge is OPEN. And occasionally it has a Boundary. Algebraic expression of the edge is
OPEN: a < 
where a is finity and represents the model edge.
You can set any value inside the disc except infinity. However, we often draw a figure which visually touches the edge. If it really does, algebraic expression should be
CLOSED: a 
.
But none of previous figures shows us any concrete scenery at infinity whether OPEN or CLOSED, though some visually touch the model edge.
No one has been to infinity. But we have storong circumstantial evidence of parallel lines as [Fig 5-2]. Here, we want to get a circumstantial evidence of Horocircle (cf. Ch.15). First, let us see how Horocircle acts in a model. To do so, we reproduced [Fig 15-2] onto what is called Poincare's Upper Half Plane (cf. Ch.21, Ch.22). That is [Fig 17-1].
Correspondence between two models is
Hyperbolic straight line in Upper Half Plane is drawn as a semicircle with the center on X-axis.
See figure (1) left. It is a assumed scenery at a point of infinity 
. It is based on [Fig 15-2] and [Fig 17-1].
[Fig 17-2] below is drawn in Klein's daisc model. Hyperbolic straight line in this model is straight. Figure (2) is based on [Fig 17-2]. Figure (1) and (2) look quite different though the object is the same. It is because that we just look models microscopically.
Consenting this, let me go on. (It would be stupid, though.)
When we use [Fig 17-2] to assume a scenery at a point of infinity , we get another scenery like figure (3) left.
If you say "It is OPEN and Keep Off there", we eliminate there. Then we get figure (4).
Considering all above, we are now able to say that
Infinity is in our hands!
To tell the truth, this lie will be exposed gradually.
Be that as it may, it is unkind that Keep Off due to OPEN.
Yet, is there any path to get the substance?
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