Is an original refinement of a theorem eligible to be published on some journal?
Generally, an extension of a theorem is valuable. But whether it is publishable is up to a publisher. The way they make a decision is to have submissions reviewed by experts. But you can, in most cases, also seek expert help in evaluating the work you do. As suggested in the comments, local professors might be good for that.
But you can just write it up and submit it. To find an appropriate journal, you need to find a few candidates and, at least, look at the index to see what sorts of things they publish.
However, it may be that your work would be considered a trivial extension rather than something with value. That would hinge on one or both of two things. First, the theorem statement itself might provide insights into understanding topological spaces. But, and probably more important, the form of the proof you provide might provide insight into the solution of other problems. Often the proof is more important than the theorem itself. Your suggestion that you needed additional lemmas to extend an old proof might be your key to success.
If/when you submit you will get feedback on the work and possibly suggestions to improve it. But it takes reading of the paper to decide on its value.