index of properties of topological spaces

Topological axioms

•
topological structures on sets
•
continuous maps
•
categories, functors and natural transformations

Compactness Properties

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compact
•
countably compact
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Lindelöf (http://planetmath.org/Lindelof), finally compact (http://planetmath.org/Lindelof)
•
locally compact
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metacompact
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paracompact
•
strongly locally compact
•
sequentially compact
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$\\sigma$ -compact (http://planetmath.org/SigmaCompact)
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$\\sigma$ -locally compact
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weakly countably compact (http://planetmath.org/LimitPointCompact), limit point compact

Countability Properties

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first countable (http://planetmath.org/FirstCountable)
•
second countable (http://planetmath.org/SecondCountable)
•
separable

Connectedness Properties

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biconnected
•
arc connected
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connected (http://planetmath.org/ConnectedSpace)
•
connected im kleinen
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extremally disconnected
•
hyperconnected
•
locally arc connected
•
locally connected
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locally path connected
•
path-connected
•
punctiform
•
scattered
•
totally disconnected
•
totally path disconnected
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totally separated
•
ultraconnected

Contractibility Proprties

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absolute retract
•
absolute neighborhood retract
•
contractible to a point

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locally simply connected
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semi-locally simply connected
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simply connected
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locally homeomorphic and covering spaces
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fibrations
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homotopy groups
•
homotopy as a functor
•
homotopy type
•
lifting properties of fibrations

Separation Properties (http://planetmath.org/SeparationAxioms)

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completely normal
•
completely regular, Tychonoff
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fully normal
•
fully $T_{4}$ (http://planetmath.org/FullyT4)
•
normal
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perfectly $T_{4}$
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perfectly normal
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regular
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semiregular
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$T_{0}$ (http://planetmath.org/T0Space), Kolmogorov (http://planetmath.org/T0Space)
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$T_{1}$ (http://planetmath.org/T1Space), Fréchet (http://planetmath.org/T1Space)
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$T_{2}$ (http://planetmath.org/T2Space), Hausdorff (http://planetmath.org/T2Space)
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$T_{2\\frac{1}{2}}$ (http://planetmath.org/CompletelyHaussdorff), completely Hausdorff
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$T_{3}$ (http://planetmath.org/SeparationAxioms)
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$T_{3\\frac{1}{2}}$ (http://planetmath.org/SeparationAxioms)
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$T_{4}$ (http://planetmath.org/SeparationAxioms)
•
$T_{5}$ (http://planetmath.org/SeparationAxioms)
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Urysohn, functionally Hausdorff

Subset Properties

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open
•
closed
•
clopen
•
regular open
•
regular closed (http://planetmath.org/RegularOpenSet)
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locally closed
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dense
•
nowhere dense
•
meager
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residual
•
separated sets (http://planetmath.org/Separated)
•
completely separated sets (http://planetmath.org/CompletelySeparated)

Topological groups

•
full subgroup

Homology

单词	IndexOfPropertiesOfTopologicalSpaces
释义	index of properties of topological spaces Topological axioms • topological structures on sets • continuous maps • categories, functors and natural transformations Compactness Properties • compact • countably compact • Lindelöf (http://planetmath.org/Lindelof), finally compact (http://planetmath.org/Lindelof) • locally compact • metacompact • paracompact • strongly locally compact • sequentially compact • $\\sigma$ -compact (http://planetmath.org/SigmaCompact) • $\\sigma$ -locally compact • weakly countably compact (http://planetmath.org/LimitPointCompact), limit point compact Countability Properties • first countable (http://planetmath.org/FirstCountable) • second countable (http://planetmath.org/SecondCountable) • separable Connectedness Properties • biconnected • arc connected • connected (http://planetmath.org/ConnectedSpace) • connected im kleinen • extremally disconnected • hyperconnected • locally arc connected • locally connected • locally path connected • path-connected • punctiform • scattered • totally disconnected • totally path disconnected • totally separated • ultraconnected Contractibility Proprties • absolute retract • absolute neighborhood retract • contractible to a point • locally simply connected • semi-locally simply connected • simply connected • locally homeomorphic and covering spaces • fibrations • homotopy groups • homotopy as a functor • homotopy type • lifting properties of fibrations Separation Properties (http://planetmath.org/SeparationAxioms) • completely normal • completely regular, Tychonoff • fully normal • fully $T_{4}$ (http://planetmath.org/FullyT4) • normal • perfectly $T_{4}$ • perfectly normal • regular • semiregular • $T_{0}$ (http://planetmath.org/T0Space), Kolmogorov (http://planetmath.org/T0Space) • $T_{1}$ (http://planetmath.org/T1Space), Fréchet (http://planetmath.org/T1Space) • $T_{2}$ (http://planetmath.org/T2Space), Hausdorff (http://planetmath.org/T2Space) • $T_{2\\frac{1}{2}}$ (http://planetmath.org/CompletelyHaussdorff), completely Hausdorff • $T_{3}$ (http://planetmath.org/SeparationAxioms) • $T_{3\\frac{1}{2}}$ (http://planetmath.org/SeparationAxioms) • $T_{4}$ (http://planetmath.org/SeparationAxioms) • $T_{5}$ (http://planetmath.org/SeparationAxioms) • Urysohn, functionally Hausdorff Subset Properties • open • closed • clopen • regular open • regular closed (http://planetmath.org/RegularOpenSet) • locally closed • dense • nowhere dense • meager • residual • separated sets (http://planetmath.org/Separated) • completely separated sets (http://planetmath.org/CompletelySeparated) Topological groups • full subgroup Homology
随便看	HomogeneousElementsOfAGradedRing homogeneous elements of a graded ring HomogeneousEquation homogeneous equation HomogeneousFunction homogeneous function HomogeneousGroup homogeneous group HomogeneousIdeal homogeneous ideal HomogeneousLinearDifferentialEquation homogeneous linear differential equation HomogeneousLinearProblem homogeneous linear problem HomogeneousPolynomial homogeneous polynomial homogeneous polynomial HomogeneousPolynomial1 HomogeneousSpace homogeneous space HomogeneousSystemOfParameters homogeneous system of parameters HomogeneousTopologicalSpace homogeneous topological space HomologicalComplexOfTopologicalVectorSpaces