Can you give me an example of a non compact topological space with compact dense subset?

I know a Hausdorff topological space $(X, \tau)$ with compact dense subset must be compact.

Hence, my intuition suggests that there may be a non compact topological space with compact dense subset.

And such space will not be Hausdorff space anymore.

But it is difficult for me to cite this example.

Please explain it in details. Thanks.