# Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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Let $d\in\mathbb N$ and $T:\mathbb R^d\to\mathbb R^d$ be bijective and continuous. I'm not sure how to approach this, but if $M\subseteq\mathbb R^d$, does $T$ map the topological boundary/interior of $... 1answer 37 views ### Does$\mathbb{Q}$have the finite-closed topology? Let$\mathbb{Q}$be the set of all rational numbers with the usual topology Does$\mathbb{Q}$have the finite-closed topology? My attempt : I think yes Finite - closed topology mean cofinite topology ... 2answers 33 views ### Rudin proof for compact subsets$\{ K_{\alpha} \}$(theorem 2.36) — Contrapositive or contradiction? I am having doubts about Theorem 2.36 pasted below. I was able to follow all the steps individually, but I don’t see how this is a proof by contradiction. It seems to be it is a proof by ... 0answers 28 views ### Show that a certain space of banded matrices with nonnegative determinants is connected and closed Let$n$and$k$be two positive integers, with$n \geq 2$and$k \geq 1$. Let$l=\{{l_i}\}$be an$nk$-dimensional positive real vector. Let$A_{n,k}(l)={(a_{i,j})}_{1 \leq i,j \leq nk}$be the ... 1answer 41 views ### How could a path be homotopic to a point I have been following Serge Lang's Intoduction to Complex Analysis at a Graduate Level and I met this theorem. I want to ask what does it mean for a function to be homotopic to a point? I am only ... 0answers 18 views ### Relationship between the co-finite topology and standard topology on R? [closed] Relationship between the co-finite and left ray and countable and standard topology on R? 0answers 25 views ### Base point of BU/BO, classifying space of U/O. There are principal bundles $$U(n) \to V_k(\mathbb{C}^n) \to G_k(\mathbb{C}^n)$$ and $$O(n) \to V_k(\mathbb{R}^n) \to G_k(\mathbb{R}^n),$$ where$V_k(\mathbb{F}^n)$and$G_k(\mathbb{F}^n)$are the ... 1answer 37 views ### Lee's Intro to Topology, generating the same topology Suppose$M$is a set and$d, d^\prime$are two different metrics on$M$. Prove that$d$, and$d'$generate the same topology on$M$if and only if the following condition is satisfied: for every$x \...
*How can every neighborhood of a limit point contain a closed set $v$ when $v$ has a complementary set in those neighborhoods ? *What is a good example of a Zariski closed dense set ? So far I have ...