I read a little remark in a topology text that the direct image of an open set under a continuous mapping is not necessarily open.
What is the definition of direct image in this case? I tried googling to no avail, and only found references to sheaf theory on wikipedia, which I don't believe to be what I'm looking for. Thanks.
They just mean "image". An example here is the image of the set $(0, 2\pi)$ under the map $\sin(x)$, which is $(0, 1]$.