Jul15 accepted Discrete topology on infinite sets Jul15 revised Discrete topology on infinite sets edited title Jul15 revised Discrete topology on infinite sets edited title Jul15 comment Discrete topology on infinite sets @ThomasRot Is the proof correct? Jul15 asked Discrete topology on infinite sets Jul2 comment precise official definition of a cell complex and CW-complex I'm reading a book on topology of surfaces. The author uses an inductive definition similar to what the O.P. stated. At the end the author says that the complex is regular. Can you please explain what does regular mean (an example if possible)? Jun30 accepted Metric on a set Jun30 comment Metric on a set I asked for a hint not a proof. Jun30 comment Metric on a set @MhenniBenghorbal Yes, that's what I need. How do I use Hagen's hint? Jun30 comment Metric on a set Yes, if we want the result to be finite. Jun30 asked Metric on a set Jun29 comment Express $\cos 6\theta$ in terms of $\cos \theta$ Use Euler's formula. Jun29 comment Continuous Functions - Topology Okay, I'll use your proof (excluding the inclusion part). So we know that $X-f^{-1}(Y-U)$ is open. But $$X-f^{-1}(Y-U)=X-(X-f^{-1}(U))=f^{-1}(U)$$. Thus $f^{-1}(U)$ is open proving that $f$ is continuous by the usual definition. Right? Jun29 comment Continuous Functions - Topology I know that a set is closed is it's complement is open. It works in the other direction as well? Jun29 accepted Continuous Functions - Topology Jun29 revised Continuous Functions - Topology deleted 73 characters in body Jun29 comment Continuous Functions - Topology Ah now I see my mistake. I'll try to fix it. Thank you guys. Jun29 comment Continuous Functions - Topology @DonAntonio We already have a theorem (or definition) that uses open sets, why can't we use it? It's a given. Jun29 comment Continuous Functions - Topology @DonAntonio That assumption is true by definition (a function is continuous if the inverse image of an open set is open). Jun29 comment Continuous Functions - Topology @DanShved No! it's the correct direction. Closeness $\implies$ Continuity