I'm going to do something rather weird here, and create a second answer. That's because I want to answer a slightly different variant of the question, and propose a controversial result.

My other answer considers the formulation of the question, and also addresses the case where the endpoints of the universe are "allowed" but are outside the universe. This one specifically considers the case where the endpoints of the universe ($x=0$ and $x=1$) are forbidden to both players.

In that case I believe (controversially)

The cop will catch the thief.

How do I arrive at this? By simply noting that every point in the universe can be visited by the cop in a finite time $t<2/3$. It doesn't matter that there are an infinite number of points to visit, the cop can still visit them all. Just like they can visit all the points between $x=0.6$ and $x=0.5$ in a tenth of a time unit. (I've neglected the parts of the universe on the right of the cop, where $x>2/3$. We know that the thief can never reach that part of the universe without being caught, because movement is continuous so they would have pass through the cop to reach it, meaning they are caught.)

If every point that the thief might exist at can be visited in finite time by the cop, the thief will be caught.

But what about the strategies that seem in indicating the thief eternally evading the cop? They are subject to the same fallacies that Zeno's paradox falls to. Zeno says that Achilles cannot catch the tortoise because there is an infinite sequence of steps for him to so so. Likewise the "thief escapes" solution says there are an infinite sequence of steps for the cop to catch the thief. Zeno is wrong because the infinite sequence of steps finishes in a finite time. These strategies are wrong for the same reason. The thief can avoid the cop for an infinite number of steps, but that does not mean he can avoid the cop for an infinite amount of time.

In my other answer I ask what happens at $t=2/3$ (with the thief following his "optimum" strategy). If we impose the restriction that no player may be at $x=0$ then the answer is unknown, but whatever answer it is it doesn't matter. The thief must be at some position $x>0$ but also with a lower x than the cop's starting point (otherwise they are caught), and the cop (as we said) can visit all those positions before $t=2/3$. If the cop can visit all the possible locations of the thief in finite time then the thief must be caught.

I'm going to do something rather weird here, and create a second answer. That's because I want to answer a slightly different variant of the question, and propose a controversial result.

My other answer considers the formulation of the question, and also addresses the case where the endpoints of the universe are "allowed" but are outside the universe. This one specifically considers the case where the endpoints of the universe ($x=0$ and $x=1$) are forbidden to both players.

In that case I believe (controversially)

The cop will catch the thief.

How do I arrive at this? By simply noting that every point in the universe can be visited by the cop in a finite time $t<2/3$. It doesn't matter that there are an infinite number of points to visit, the cop can still visit them all. Just like they can visit all the points between $x=0.6$ and $x=0.5$ in a tenth of a time unit, even though that is an infinite number of points. (I've neglected the parts of the universe on the right of the cop, where $x>2/3$. We know that the thief can never reach that part of the universe without being caught, because movement is continuous so they would have pass through the cop to reach it, meaning they are caught.)

If every point that the thief might exist at can be visited in finite time by the cop, the thief will be caught.

But what about the strategies that seem in indicating the thief eternally evading the cop? They are subject to the same fallacies that Zeno's paradox falls to. Zeno says that Achilles cannot catch the tortoise because there is an infinite sequence of steps for him to so so. Likewise the "thief escapes" solution says there are an infinite sequence of steps for the cop to catch the thief. Zeno is wrong because the infinite sequence of steps finishes in a finite time. These strategies are wrong for the same reason. The thief can avoid the cop for an infinite number of steps, but that does not mean he can avoid the cop for an infinite amount of time.

In my other answer I ask what happens at $t=2/3$ (with the thief following his "optimum" strategy). If we impose the restriction that no player may be at $x=0$ then the answer is unknown, but whatever answer it is it doesn't matter. The thief must be at some position $x>0$ but also with a lower x than the cop's starting point (otherwise they are caught), and the cop (as we said) can visit all those positions before $t=2/3$. If the cop can visit all the possible locations of the thief in finite time then the thief must be caught.

I'm going to do something rather weird here, and create a second answer. That's because I want to answer a slightly different variant of the question, and propose a controversial result.

My other answer considers the formulation of the question, and also addresses the case where the endpoints of the universe are "allowed" but are outside the universe. This one specifically considers the case where the endpoints of the universe ($x=0$ and $x=1$) are forbidden to both players.

In that case I believe (controversially)

The cop will catch the thief.

How do I arrive at this? By simply noting that every point in the universe can be visited by the cop in a finite time $t<2/3$. It doesn't matter that there are an infinite number of points to visit, the cop can still visit them all. Just like they can visit all the points between $x=0.6$ and $x=0.5$ in a tenth of a time unit. (I've neglected the parts of the universe on the right of the cop, where $x>2/3$. We know that the thief can never reach that part of the universe without being caught, because movement is continuous so they would have pass through the cop to reach it, meaning they are caught.)

If every point that the thief might exist at can be visited in finite time by the cop, the thief will be caught.

But what about the strategies that seem in indicating the thief eternally evading the cop? They are subject to the same fallacies that Zeno's paradox falls to. Zeno says that Achilles cannot catch the tortoise because there is an infinite sequence of steps for him to so so. Likewise the "thief escapes" solution says there are an infinite sequence of steps for the cop to catch the thief. Zeno is wrong because the infinite sequence of steps finishes in a finite time. These strategies are wrong for the same reason. The thief can avoid the cop for an infinite number of steps, but that does not mean he can avoid the cop for an infinite amount of time.

In my other answer I ask what happens at $t=2/3$ (with the thief following his "optimum" strategy). If we impose the restriction that no player may be at $x=0$ then the answer is unknown, but whatever answer it is it doesn't matter. The thief must be at some position $x>0$ but also with a lower x than the cop (otherwise they are caught), and the cop (as we said) can visit all those positions before $t=2/3$.

I'm going to do something rather weird here, and create a second answer. That's because I want to answer a slightly different variant of the question, and propose a controversial result.

My other answer considers the formulation of the question, and also addresses the case where the endpoints of the universe are "allowed" but are outside the universe. This one specifically considers the case where the endpoints of the universe ($x=0$ and $x=1$) are forbidden to both players.

In that case I believe (controversially)

The cop will catch the thief.

How do I arrive at this? By simply noting that every point in the universe can be visited by the cop in a finite time $t<2/3$. It doesn't matter that there are an infinite number of points to visit, the cop can still visit them all. Just like they can visit all the points between $x=0.6$ and $x=0.5$ in a tenth of a time unit. (I've neglected the parts of the universe on the right of the cop, where $x>2/3$. We know that the thief can never reach that part of the universe without being caught, because movement is continuous so they would have pass through the cop to reach it, meaning they are caught.)

If every point that the thief might exist at can be visited in finite time by the cop, the thief will be caught.

But what about the strategies that seem in indicating the thief eternally evading the cop? They are subject to the same fallacies that Zeno's paradox falls to. Zeno says that Achilles cannot catch the tortoise because there is an infinite sequence of steps for him to so so. Likewise the "thief escapes" solution says there are an infinite sequence of steps for the cop to catch the thief. Zeno is wrong because the infinite sequence of steps finishes in a finite time. These strategies are wrong for the same reason. The thief can avoid the cop for an infinite number of steps, but that does not mean he can avoid the cop for an infinite amount of time.

In my other answer I ask what happens at $t=2/3$ (with the thief following his "optimum" strategy). If we impose the restriction that no player may be at $x=0$ then the answer is unknown, but whatever answer it is it doesn't matter. The thief must be at some position $x>0$ but also with a lower x than the cop's starting point (otherwise they are caught), and the cop (as we said) can visit all those positions before $t=2/3$. If the cop can visit all the possible locations of the thief in finite time then the thief must be caught.