Is a Covering Space of a Topological Space always Hausdorff?

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Is a Covering Space of a Topological Space always Hausdorff?

I can separate two different points from the same fiber, but what about two arbitrary points?

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A covering space of a Hausdorff space is always Hausdorff. Either you can separate points in the base and take preimages, or you can use your proof if the points lie in the same fiber.