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, neighbourhood, boundary, first and second countability, separablility are toological properties. We shall come across several topological properties in the sequel. Because of its critical role the subject topology is usually described as the study of topological properties.

• Let X = \left] { - 1,1} \right[ and f:X \to \mathbb{R} 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 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 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.