Open sets, bases, continuity, connectedness, compactness, Hausdorff spaces, homeomorphisms and the fundamental group.
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- What is an open set in a topological space (X, τ)?
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A set U ⊆ X such that U ∈ τ, where τ is the topology on X.- State the property: the union of any collection of open sets is
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Open.- State the property: the intersection of finitely many open sets is
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Open.- What is the complement of an open set?
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A closed set.- Are the empty set and the full space open in any topology?
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Yes, both are always open by definition of a topology.- What is a basis (or base) for a topology on X?
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A collection B of open sets such that every open set is a union of sets from B.- State the basis characterization theorem: a collection B is a basis for a topology iff
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Every point in X is in some B ∈ B, and if x ∈ B1 ∩ B2 for B1, B2 ∈ B, then x ∈ B3 ⊆ B1 ∩ B2 for some B3 ∈ B.- What is a subbasis for a topology?
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A collection S of subsets such that the collection of all finite intersections of sets in S forms a basis.- What is the standard basis for the usual topology on ℝ?
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The collection of all open intervals (a, b) where a < b.- When is a function f : X → Y continuous between topological spaces?
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When the preimage of every open set in Y is open in X.- Define continuity of f at a point x ∈ X using neighborhoods.
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For every neighborhood V of f(x), there exists a neighborhood U of x such that f(U) ⊆ V.- Is the composition of two continuous functions continuous?
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Yes, if f : X → Y and g : Y → Z are continuous, then g ∘ f : X → Z is continuous.- Is the identity function id : X → X always continuous?
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Yes, in any topology on X.- What is a connected space?
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A topological space that cannot be written as the union of two disjoint nonempty open sets.- Is the continuous image of a connected space connected?
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Yes, if f : X → Y is continuous and X is connected, then f(X) is connected.
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