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Analytic Topology Definitions

State the write term for each of these definitions in analytic topology.
Quiz by josh100
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Last updated: April 21, 2022
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First submittedApril 21, 2022
Times taken2
Average score80.8%
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Hint
Answer
T1 and normal
t4
For all x ∈ X, for all open U ∋ x, there
exists connected C and open V such that x ∈ V ⊆ C ⊆ U
locally connected
For all x ∈ X and open U ∋ x, there exists open V and compact K such that x ∈ V ⊆ K ⊆ U
locally compact
There is continuous f : X → [0, 1] such that f(A) ⊆ {0} and f(B) ⊆ {1
functionally separated
T is the coarsest topology such that all πλ are continuous
tychonoff topology
Every open cover has a locally finite open refinement
paracompact
∀u ∈ U, ∃v ∈ V, u ⊆ v
refinement
For all x ∈ X, there exists open V ∋ x such that V ∩ U /= ∅ for only finitely many U ∈ U
locally finite
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
Every countable open cover
has a finite subcover
countably compact
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
For all A ⊆ X, either A ∈ U, or X \ A ∈ U
ultafilter
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
Hint
Answer
f is closed, and has compact fibres
proper
Whenever C is a closed set, and x /∈ C, {x} and C are functionally separated
completely regular
The set of all F ⊆ X such that x ∈ F◦
neighbourhood filter
X has a countable basis
second countable
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
T1 and regular
t3
1. ≤ is a total order of Y , and
2. Every non-empty subset of Y has a ≤-least element
well ordering
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
Completely regular and T1
tychonoff
Every open cover has a countable
subcover
lindelof
Whenever x /= y ∈ X, there exists open U,
V such that x ∈ U, y ∈ V and U ∩ V = ∅
t2
Whenever x and y are distinct points
of X, there is an open set U such that x ∈ U and y /∈ U
t1
Compact Hausdorff space with a basis of clopen
sets
stone space
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