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Homework Statement
Consider the sets [itex]A=\{(t,\sin(1/t))\in \mathbb{R}^2:t\in(0,1]\}[/itex], [itex]B=\{(0,s)\in\mathbb{R}^2:s\in[-1,1]\}[/itex]. Let [itex]X=A\cup B[/itex]. We consider on X the topology induced by the open ball topology of R².
a) Is X connected?
b) Is X path connected?
The Attempt at a Solution
a) I found that it is connected.
b) I concluded that it is not path-connected but it's a little touchy and I want a second opinion.
In case you haven't visualized the set yet, it consists of the union of the vertical line [itex]\{0\} \times [-1,1][/itex] with a sine wave on (0,1] that oscillates faster and faster as t-->0.
There is of course no difficulty in connecting two points of A or two points of B. The interesting case is when one tries to connect a point of A with one of B. Without loss of generality, let's try to connect (1,sin(1)) with (0,0). The only way is to follow the sine wave:
[tex]\gamma:[0,1]\rightarrow X[/itex]
[tex]\gamma(t)=\left\{\begin{array}{cc}(0,0)&\mbox{if} \ \
t=0\\(t,\sin(1/t)) & \mbox{if} \ \ t\in(0,1]\end{array}[/tex]
Is gamma continuous? Let's consider an open ball of radius, say, ½, centered at (0,0). The intersection of that ball with X is an open in X that we will call O. Let's look at the pre-image of O by gamma. It contains {0} and an infinity of open intervals of (0,1] but [itex]\gamma^{-1}(O)[/itex] is not open because it is impossible to find an open nbh of {0} that be entirely in [itex]\gamma^{-1}(O)[/itex].
N.B. We haven't covered caracterisation of continuity in metrizable spaces, so I can't use that.
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