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Related concepts
References
A topological property is a property/predicate of topological spaces which is invariant under isomorphism of topological spaces, hence under homeomorphism: homeomorphism invariant.
uniform property
epsilontic analysis
homotopy invariant
References
- Wikipedia, Topological property
A property $$P$$ is said to be a topological property if whenever a space $$X$$ has the property $$P$$, all spaces which are homeomorphic to $$X$$ also have the property $$P$$, $$X simeq Y simeq Z$$.
In other words, a topological property is a property which, if possessed by a topological space, is also possessed by all topological spaces homeomorphic to that space.
Note: It may noted that length, angle, boundedness, Cauchy sequence, straightness and being triangular or circular are not topological properties, whereas limit point, interior, neighborhood, boundary, first and second countability, and separablility are topological properties. We shall come across several topological properties in a following post. Because of its critical role the subject topology, it is usually described as the study of topological properties.
Examples:
• Let $$X = left] – 1,1 right[$$ and $$f:X to mathbbR$$ be defined by $$fleft( x right) = tan left( fracpi x2 right)$$. Then $$f$$ is a homeomorphism and therefore $$left] – 1,1 right[ simeq mathbbR$$. Note that $$left] – 1,1 right[$$ and $$mathbbR$$ have different lengths, therefore length is not a topological property. Also $$X$$ is bounded and $$mathbbR$$ is not bounded, therefore boundeness is not a topological property.
• Let $$f:left] 0,infty right[ to left] 0,infty right[$$ defined by $$fleft( x right) = frac1x$$, then $$f$$ is a homeomorphism. Consider the sequences $$left( x_n right) = left( 1,frac12,frac13, cdots right)$$ and $$left( fleft( x_n right) right) = left( 1,2,3, ldots right)$$ in $$left] 0,infty right[$$. $$left( x_n right)$$ is a Cauchy sequence, where $$left( fleft( x_n right) right)$$ is not. Therefore, being a Cauchy sequence is not a topological property.
• Straightness is not a topological property, for a line may be bent and stretched until it is wiggly.
• Being triangular is not a topological property since a triangle can be continuously deformed into a circle and conversely.
We can formalise topology using only the language of set theory. [For instance, a topological space is a pair $langle X, tau rangle$ where $X$ and $tau$ are sets satisfying various properties, and we can define a homeomorphism $langle X, tau rangle to langle Y, sigma rangle$ as a function $X to Y$ (which is itself a set) satisfying some conditions, etc. All this can be formalised.]
So we can define a unary predicate $textTS$ defined by
$$forall x[textTS(x) leftrightarrow x textis a topological space]$$
where ‘$x textis a topological space$’ is shorthand for…
$$beginalignexists X exists tau( hspace53pt\
x= langle X, tau rangle wedge &tau subseteq mathcalP(X) wedge varnothing in tau wedge X in tau\
wedge & forall U forall V [U in tau wedge V in tau to U cap V in tau]\
wedge & forall A [A subseteq tau to bigcup A in tau]\
) hspace78pt endalign$$
Now suppose $phi$ is a formula with one không lấy phí variable, $x$ say. Then $phi$ is a topological property (i.e. is preserved under homeomorphism) if
$$forall x [textTS(x) wedge phi(x) rightarrow forall y[textTS(y) wedge x cong y rightarrow phi(y)]]$$
That is, if $phi$ holds for any space $x$ then for any space $y$ homeomorphic to $x$, $phi$ holds for $y$.
Here I’ve used $x cong y$ as shorthand for the formula expressing that $x=langle X, tau rangle$ and $y=langle Y, sigma rangle$ are homeomorphic.
Is this what you were after?
Frankly, I don’t see how it’s any more enlightening to put yourself through all this than it is to just say “a topological property is one that is preserved by homeomorphism”, as so succinctly put by Thomas Andrews in the comments.
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are not homeomorphic, it is sufficient to find a topological property which is not shared by them.
- The cardinality |X| of the space X.
The cardinality
|
displaystyle vert
τ(X)
|
displaystyle vert
of the topology of the space X.
Weight w(X), the least cardinality of a basis of the topology of the space X.
Density d(X), the least cardinality of a subset of X whose closure is X.
Note that some of these terms are defined differently in older mathematical literature; see history of the separation axioms.
- T0 or Kolmogorov. A space is Kolmogorov if for every pair of distinct points x and y in the space, there is least either an open set containing x but not y, or an open set containing y but not x.
T1 or Fréchet. A space is Fréchet if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T0; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T1 if all its singletons are closed. T1 spaces are always T0.
Sober. A space is sober if every irreducible closed set C has a unique generic point p. In other words, if C is not the (possibly nondisjoint) union of two smaller closed subsets, then there is a p such that the closure of p equals C, and p is the only point with this property.
T2 or Hausdorff. A space is Hausdorff if every two distinct points have disjoint neighbourhoods. T2 spaces are always T1.
T2½ or Urysohn. A space is Urysohn if every two distinct points have disjoint closed neighbourhoods. T2½ spaces are always T2.
Completely T2 or completely Hausdorff. A space is completely T2 if every two distinct points are separated by a function. Every completely Hausdorff space is Urysohn.
Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods.
T3 or Regular Hausdorff. A space is regular Hausdorff if it is a regular T0 space. (A regular space is Hausdorff if and only if it is T0, so the terminology is consistent.)
Completely regular. A space is completely regular if whenever C is a closed set and p is a point not in C, then C and p are separated by a function.
T3½, Tychonoff, Completely regular Hausdorff or Completely T3. A Tychonoff space is a completely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.
Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity.
T4 or Normal Hausdorff. A normal space is Hausdorff if and only if it is T1. Normal Hausdorff spaces are always Tychonoff.
Completely normal. A space is completely normal if any two separated sets have disjoint neighbourhoods.
T5 or Completely normal Hausdorff. A completely normal space is Hausdorff if and only if it is T1. Completely normal Hausdorff spaces are always normal Hausdorff.
Perfectly normal. A space is perfectly normal if any two disjoint closed sets are precisely separated by a function. A perfectly normal space must also be completely normal.
T6 or Perfectly normal Hausdorff, or perfectly T4. A space is perfectly normal Hausdorff, if it is both perfectly normal and T1. A perfectly normal Hausdorff space must also be completely normal Hausdorff.
Discrete space. A space is discrete if all of its points are completely isolated, i.e. if any subset is open.
Number of isolated points. The number of isolated points of a topological space.
Countability conditions
- Separable. A space is separable if it has a countable dense subset.
First-countable. A space is first-countable if every point has a countable local base.
Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.
Connectedness
- Connected. A space is connected if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only clopen sets are the empty set and itself.
Locally connected. A space is locally connected if every point has a local base consisting of connected sets.
Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point.
Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected.
Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.
Arc-connected. A space X is arc-connected if for every two points x, y in X, there is an arc f from x to y, i.e., an injective continuous map f: [0,1] → X with p(0) = x and p(1) = y. Arc-connected spaces are path-connected.
Simply connected. A space X is simply connected if it is path-connected and every continuous map f: S1 → X is homotopic to a constant map.
Locally simply connected. A space X is locally simply connected if every point x in X has a local base of neighborhoods U that is simply connected.
Semi-locally simply connected. A space X is semi-locally simply connected if every point has a local base of neighborhoods U such that every loop in U is contractible in X. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a universal cover.
Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected.
Hyperconnected. A space is hyperconnected if no two non-empty open sets are disjoint. Every hyperconnected space is connected.
Ultraconnected. A space is ultraconnected if no two non-empty closed sets are disjoint. Every ultraconnected space is path-connected.
Indiscrete or trivial. A space is indiscrete if the only open sets are the empty set and itself. Such a space is said to have the trivial topology.
Compactness
- Compact. A space is compact if every open cover has a finite subcover. Some authors call these spaces quasicompact and reserve compact for Hausdorff spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.
Sequentially compact. A space is sequentially compact if every sequence has a convergent subsequence.
Countably compact. A space is countably compact if every countable open cover has a finite subcover.
Pseudocompact. A space is pseudocompact if every continuous real-valued function on the space is bounded.
σ-compact. A space is σ-compact if it is the union of countably many compact subsets.
Lindelöf. A space is Lindelöf if every open cover has a countable subcover.
Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.
Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact Hausdorff spaces are always Tychonoff.
Ultraconnected compact. In an ultra-connected compact space X every open cover must contain X itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a monolith.
Metrizability
- Metrizable. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Moreover, a topological space (X,T) is said to be metrizable if there exists a metric for X such that the metric topology T(d) is identical with the topology T.
Polish. A space is called Polish if it is metrizable with a separable and complete metric.
Locally metrizable. A space is locally metrizable if every point has a metrizable neighbourhood.
Miscellaneous
- Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense.
Door space. A topological space is a door space if every subset is open or closed (or both).
Topological Homogeneity. A space X is (topologically) homogeneous if for every x and y in X there is a homeomorphism
f
:
X
→
X
displaystyle f:Xto X
such that
f
(
x
)
=
y
.
displaystyle f(x)=y.
Intuitively speaking, this means that the space looks the same every point. All topological groups are homogeneous.
Finitely generated or Alexandrov. A space X is Alexandrov if arbitrary intersections of open sets in X are open, or equivalently if arbitrary unions of closed sets are closed. These are precisely the finitely generated members of the category of topological spaces and continuous maps.
Zero-dimensional. A space is zero-dimensional if it has a base of clopen sets. These are precisely the spaces with a small inductive dimension of 0.
Almost discrete. A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
Boolean. A space is Boolean if it is zero-dimensional, compact and Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the Stone spaces of Boolean algebras.
Reidemeister torsion
κ
displaystyle kappa
-resolvable. A space is said to be κ-resolvable[1] (respectively: almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (respectively: almost disjoint over the ideal of nowhere dense subsets). If the space is not
κ
displaystyle kappa
-resolvable then it is called
κ
displaystyle kappa
-irresolvable.
Maximally resolvable. Space
X
displaystyle X
is maximally resolvable if it is
Δ
(
X
)
displaystyle Delta (X)
-resolvable, where
Δ
(
X
)
=
min
.
displaystyle Delta (X)=min.
Number
Δ
(
X
)
displaystyle Delta (X)
is called dispersion character of
X
.
displaystyle X.
Strongly discrete. Set
D
displaystyle D
is strongly discrete subset of the space
X
displaystyle X
if the points in
D
displaystyle D
may be separated by pairwise disjoint neighborhoods. Space
X
displaystyle X
is said to be strongly discrete if every non-isolated point of
X
displaystyle X
is the accumulation point of some strongly discrete set.
There are many examples of properties of metric spaces, etc, which are not topological properties. To show a property
P
displaystyle P
is not topological, it is sufficient to find two homeomorphic topological spaces
X
≅
Y
displaystyle Xcong Y
such that
X
displaystyle X
has
P
displaystyle P
, but
Y
displaystyle Y
does not have
P
displaystyle P
.
For example, the metric space properties of boundedness and completeness are not topological properties. Let
X
=
R
displaystyle X=mathbb R
and
Y
=
(
−
π
2
,
π
2
)
displaystyle Y=(-tfrac pi 2,tfrac pi 2)
be metric spaces with the standard metric. Then,
X
≅
Y
displaystyle Xcong Y
via the homeomorphism
arctan
:
X
→
Y
displaystyle operatorname arctan colon Xto Y
. However,
X
displaystyle X
is complete but not bounded, while
Y
displaystyle Y
is bounded but not complete.
This article is in list format but may read better as prose. You can help by converting this article, if appropriate. Editing help is available. (March 2022)
- Euler characteristic
Winding number
Characteristic class
Characteristic numbers
Chern class
Knot invariant
Linking number
Fixed-point property
Topological quantum number
Homotopy group and Cohomotopy group
Homology and cohomology
Quantum invariant
^ Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán (2008). “Resolvability and monotone normality”. Israel Journal of Mathematics. 166 (1): 1–16. arXiv:math/0609092. doi:10.1007/s11856-008-1017-y. ISSN 0021-2172. S2CID 14743623.
- Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. p. 369. ISBN 9780486434797.
Munkres, James R. (2000). Topology. Prentice-Hall. ISBN 0-13-181629-2.
[2] Simon Moulieras, Maciej Lewenstein and Graciana Puentes, Entanglement engineering and topological protection by discrete-time quantum walks, Journal of Physics B: Atomic, Molecular and Optical Physics 46 (10), 104005 (2013).
https://iopscience.iop.org/article/10.1088/0953-4075/46/10/104005/pdf
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#area #topological #property – 2022-03-12 08:47:10