General Topology
Read this topicExplanations 1
Problems 10
- Let FFF be a subspace of a topological space XXX. How can you tell if FFF is irreducible? Select all that apply.
- Select all properties of irreducible topological spaces XXX. (Source: Cf. Qing Liu, Chapter 2, Proposition 4.5)
- Let XXX be a topological space and Y={ξ}Y = \{\xi\}Y={ξ} a singleton subspace of XXX. (Here ξ\xiξ is just a point of XXX.) When does a point p∈Xp\in Xp∈X lie in the closure Y‾\o…
- If a continuous map f :X→Yf\colon X\to Yf:X→Y is both open and closed, then what would guarantee that fff is surjective? Suppose XXX is nonempty.
- If UUU and CCC are open and closed subsets of XXX, respectively, what can you say about the relative containment of p,ξp,\xip,ξ in UUU and CCC? Mark all true implications.
- Can you cover the real plane, R2\mathbb{R}^2R2, using a disjoint set of topological T's? \\ [Source: Communicated by Mark Sellke, and separately by David (Ting-Chun) Lin, who he…
- Can you cover the real plane, R2\mathbb{R}^2R2, using a disjoint set of topological O's? \\ [Source: Classical?]
- What is the data of a morphism G→π∗F\mathcal{G} \to \pi_\ast \mathcal{F}G→π∗F of sheaves over YYY equivalent to? Select all that apply.
- Which of the following conditions are equivalent to ϕ\phiϕ being a local homomorphism? Select all that apply.
- If p :E→Bp\colon E\to Bp:E→B is a covering map such that BBB is simply connected, must EEE be simply connected?