Is area a topological property
A topological property is a property/predicate of topological spaces which is invariant under isomorphism of topological spaces, hence under homeomorphism: homeomorphism invariant.
References
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 $$f:\left] {0,\infty } \right[ \to \left] {0,\infty } \right[$$ defined by $$f\left( x \right) = \frac{1}{x}$$, then $$f$$ is a homeomorphism. Consider the sequences $$\left( {{x_n}} \right) = \left( {1,\frac{1}{2},\frac{1}{3}, \cdots } \right)$$ and $$\left( {f\left( {{x_n}} \right)} \right) = \left( {1,2,3, \ldots } \right)$$ in $$\left] {0,\infty } \right[$$. $$\left( {{x_n}} \right)$$ is a Cauchy sequence, where $$\left( {f\left( {{x_n}} \right)} \right)$$ 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 $\text{TS}$ defined by $$\forall x[\text{TS}(x) \leftrightarrow x\ \text{is a topological space}]$$ where '$x\ \text{is a topological space}$' is shorthand for... $$\begin{align}\exists X \exists \tau( \hspace{53pt}\\ x= \langle X, \tau \rangle \wedge &\tau \subseteq \mathcal{P}(X) \wedge \varnothing \in \tau \wedge X \in \tau\\ \wedge & \forall U\ \forall V\ [U \in \tau \wedge V \in \tau \to U \cap V \in \tau]\\ \wedge & \forall A\ [A \subseteq \tau \to \bigcup A \in \tau]\\ ) \hspace{78pt} \end{align}$$ Now suppose $\phi$ is a formula with one free variable, $x$ say. Then $\phi$ is a topological property (i.e. is preserved under homeomorphism) if $$\forall x [\text{TS}(x) \wedge \phi(x) \rightarrow \forall y[\text{TS}(y) \wedge x \cong y \rightarrow \phi(y)]]$$ 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.
Main article: Cardinal function § Cardinal functions in topology Main article: Separation axiom Note that some of these terms are defined differently in older mathematical literature; see history of the separation axioms.
See also: Axiom of countability 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 X\cong 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 X\cong Y}
via the homeomorphism
arctan
:
X
→
Y
{\displaystyle \operatorname {arctan} \colon X\to Y}
. However,
X
{\displaystyle X}
is complete but not bounded, while
Y
{\displaystyle Y}
is bounded but not complete.
[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 Retrieved from "https://en.wikipedia.org/w/index.php?title=Topological_property&oldid=1067004221" |