I want to see a plot of closed time-like curve in $(t,x)$.
$t$ - vertical axis
$x$ horizontal axis
(the usual setting just neglect $y$ and $z$ components of $(t,x,y,z)$).
What does it look like?
You cannot look at a curve through space-time and say whether or not it is time-like without considering the metric. At each point in space-time the metric defines the boundary of the local increments that are time-like. Another name for this boundary is the light cone. If, at each point along the curve, the tangent of the curve lies inside the light cone at that point, then the curve is time-like. If the curve is also closed, then it is a closed time-like curve. For solutions of Einstein’s equation with massive bodies, the metric gives rise to a warped space-time, and for any choice of coordinates the light cones will be different from point to point. But for ordinary cases, there will still not be closed time-like curves. However, for some solutions, such as within the event horizon of a spinning black hole, there are closed time-like curves.
So, a short answer to the question is draw a closed curve on the plane. Then draw little light cones along the curve. Since there are only two dimensions, each light cone looks like an x, and you can shade two opposing regions to indicate the forward and backward time-like regions of each x, which can be labeled + and -. The curve should consistently go through the center of each x from - to +, and not go through the unshaded (space-like) regions of any x. Then you can say you have a time-like curve. But without all those local light-cones, you can’t say one way or the other whether your closed curve is time-like.
Also, it would then be misleading to label the axes x and t, since you now just have general coordinates, and globally, neither one is completely time-like or space-like.