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 $f:X \to \mathbb{R}$ be defined by $f\left( x \right) = \tan \left( {\frac{{\pi x}}{2}} \right)$. Then $f$ is a homeomorphism and therefore $\left] { - 1,1} \right[ \simeq \mathbb{R}$. Note that $\left] { - 1,1} \right[$ and $\mathbb{R}$ have different lengths, therefore length is not a topological property. Also $X$ is bounded and $\mathbb{R}$ is not bounded, therefore boundeness is not a topological property.

• 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.