Statistics for Analytic Topology Definitions

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General Stats

This quiz has been taken 2 times
The average score is 21 of 26

Answer Stats

Hint	Answer	% Correct
Whenever C is a closed set, and x /∈ C, {x} and C are functionally separated	completely regular	100%
Every countable open cover has a finite subcover	countably compact	100%
There is continuous f : X → [0, 1] such that f(A) ⊆ {0} and f(B) ⊆ {1	functionally separated	100%
h : X → Y has the following properties: a) h is one-to-one, b) h is a homeomorphism from X to h(X), c) h(X) is dense in Y	hausdorff compactification	100%
Every open cover has a countable subcover	lindelof	100%
For all x ∈ X and open U ∋ x, there exists open V and compact K such that x ∈ V ⊆ K ⊆ U	locally compact	100%
For all x ∈ X, for all open U ∋ x, there exists connected C and open V such that x ∈ V ⊆ C ⊆ U	locally connected	100%
The set of all F ⊆ X such that x ∈ F◦	neighbourhood filter	100%
Whenever C and D are disjoint and closed in X, there exist open U and V such that C ⊆ U, D ⊆ V , and U ∩ V = ∅	normal	100%
Every open cover has a locally finite open refinement	paracompact	100%
f is closed, and has compact fibres	proper	100%
∀u ∈ U, ∃v ∈ V, u ⊆ v	refinement	100%
Whenever x ∈ X and C ⊆ X is closed, and x /∈ C, there exist open U, V such that x ∈ U, C ⊆ V and U ∩ V = ∅	regular	100%
X has a countable basis	second countable	100%
Whenever x and y are distinct points of X, there is an open set U such that either x ∈ U and y /∈ U, or x /∈ U and y ∈ U	t0	100%
Whenever x and y are distinct points of X, there is an open set U such that x ∈ U and y /∈ U	t1	100%
Whenever x /= y ∈ X, there exists open U, V such that x ∈ U, y ∈ V and U ∩ V = ∅	t2	100%
T1 and regular	t3	100%
T1 and normal	t4	100%
Completely regular and T1	tychonoff	100%
T is the coarsest topology such that all πλ are continuous	tychonoff topology	100%
1. αX is a topological space with points X ∪ {∗}, such that U is open iff a) U is open in X, or b) ∗ ∈ U, and X \ U is compact. 2. h : X → X ∪ {∗} is the identity	alexandroff compatification	0%
For all x ∈ X, there exists open V ∋ x such that V ∩ U /= ∅ for only finitely many U ∈ U	locally finite	0%
Compact Hausdorff space with a basis of clopen sets	stone space	0%
For all A ⊆ X, either A ∈ U, or X \ A ∈ U	ultafilter	0%
1. ≤ is a total order of Y , and 2. Every non-empty subset of Y has a ≤-least element	well ordering	0%