Homeomorphism: Difference between revisions

Latest revision as of 11:39, 15 February 2025

Homeomorphism[edit]

In the field of topology, a homeomorphism is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces; they are the mappings that preserve all the topological properties of a space.

A homeomorphism can be thought of as a "stretching" or "bending" of a space into another without tearing or gluing. Two spaces that are homeomorphic are considered to be topologically equivalent.

Definition[edit]

Let \(X\) and \(Y\) be topological spaces. A function \(f: X \to Y\) is a homeomorphism if it satisfies the following conditions:

Bijective: \(f\) is a bijective function, meaning it is both injective and surjective.
Continuous: \(f\) is a continuous function.
Inverse is continuous: The inverse function \(f^{-1}: Y \to X\) is also continuous.

If such a function exists, \(X\) and \(Y\) are said to be homeomorphic, and we write \(X \cong Y\).

Examples[edit]

The circle \(S^1\) is homeomorphic to any ellipse in the plane. This is because an ellipse can be continuously deformed into a circle without cutting or gluing.
The surface of a sphere is homeomorphic to the surface of any ellipsoid.
The Möbius strip is not homeomorphic to a cylinder, as they have different topological properties.

Properties[edit]

Homeomorphisms preserve topological properties such as:

Connectedness: If a space is connected, any space homeomorphic to it is also connected.
Compactness: A compact space remains compact under a homeomorphism.
Hausdorff property: If a space is Hausdorff, any space homeomorphic to it is also Hausdorff.

Applications[edit]

Homeomorphisms are fundamental in topology because they allow mathematicians to classify spaces based on their topological properties rather than their geometric shape. This concept is crucial in areas such as:

Algebraic topology, where spaces are studied up to homeomorphism.
Differential topology, where smooth structures are considered up to diffeomorphism, a type of homeomorphism.
Knot theory, where knots are studied as embeddings of circles in 3-dimensional space, up to ambient isotopy, a form of homeomorphism.

Related pages[edit]

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-'''Homeomorphism''' is a fundamental concept in the field of [[topology]], which is a major area of study in [[mathematics]]. It refers to a special kind of [[function]] between two [[topological space]]s that establishes a correspondence between the spaces that is both continuous and invertible, with the inverse function also being continuous. This concept is crucial for understanding the intrinsic properties of topological spaces that remain unchanged under continuous deformations, such as stretching or bending, but not tearing or gluing.
+== Homeomorphism ==
-==Definition==
+[[File:Blue_Trefoil_Knot.png|thumb|right|A blue trefoil knot, an example of a topological space.]]
-Formally, a function ''f'' : ''X'' → ''Y'' between two topological spaces ''X'' and ''Y'' is called a homeomorphism if it satisfies three conditions:
-# ''f'' is a [[bijection]] (one-to-one and onto),
-# ''f'' is [[continuous function|continuous]], and
-# The inverse function ''f''<sup>−1</sup> is continuous.
-When such a function exists, the spaces ''X'' and ''Y'' are said to be homeomorphic. This relationship is denoted as ''X'' ≈ ''Y''.
+In the field of [[topology]], a '''homeomorphism''' is a continuous function between two [[topological spaces]] that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces; they are the mappings that preserve all the topological properties of a space.
-==Properties==
+A homeomorphism can be thought of as a "stretching" or "bending" of a space into another without tearing or gluing. Two spaces that are homeomorphic are considered to be topologically equivalent.
-Homeomorphic spaces share many topological properties, such as:
-* [[Compact space|Compactness]]
-* [[Connected space|Connectedness]]
-* [[Dimension (topology)|Dimension]]
-* [[Fundamental group]]
-These shared properties make homeomorphism a key concept in classifying and studying topological spaces.
+== Definition ==
-==Examples==
+Let \(X\) and \(Y\) be topological spaces. A function \(f: X \to Y\) is a '''homeomorphism''' if it satisfies the following conditions:
-. The surface of a donut and a coffee cup are homeomorphic, as each can be deformed into the shape of the other without cutting or gluing. This is a popular example used to illustrate the concept of homeomorphism in an intuitive way.
-. The real number line ''R'' and any open interval ''(a, b)'' in ''R'' are homeomorphic, as they can be related by a continuous, bijective function with a continuous inverse.
-==Applications==
+# '''Bijective''': \(f\) is a [[bijective function]], meaning it is both [[injective]] and [[surjective]].
-Homeomorphism plays a crucial role in various areas of mathematics and its applications, including:
+# '''Continuous''': \(f\) is a [[continuous function]].
-* [[Differential geometry]]
+# '''Inverse is continuous''': The inverse function \(f^{-1}: Y \to X\) is also continuous.
-* [[Algebraic topology]]
-* [[Geometric topology]]
-* [[Dynamical systems]]
-It helps in understanding the intrinsic geometry of objects and spaces, and in classifying spaces according to their topological properties.
+If such a function exists, \(X\) and \(Y\) are said to be '''homeomorphic''', and we write \(X \cong Y\).
-==See Also==
+== Examples ==
-* [[Topology]]
+* The [[circle]] \(S^1\) is homeomorphic to any [[ellipse]] in the plane. This is because an ellipse can be continuously deformed into a circle without cutting or gluing.
+* The [[surface]] of a [[sphere]] is homeomorphic to the surface of any [[ellipsoid]].
+* The [[Möbius strip]] is not homeomorphic to a [[cylinder]], as they have different topological properties.
+== Properties ==
+Homeomorphisms preserve topological properties such as:
+* [[Connectedness]]: If a space is connected, any space homeomorphic to it is also connected.
+* [[Compactness]]: A compact space remains compact under a homeomorphism.
+* [[Hausdorff space|Hausdorff property]]: If a space is Hausdorff, any space homeomorphic to it is also Hausdorff.
+== Applications ==
+Homeomorphisms are fundamental in [[topology]] because they allow mathematicians to classify spaces based on their topological properties rather than their geometric shape. This concept is crucial in areas such as:
+* [[Algebraic topology]], where spaces are studied up to homeomorphism.
+* [[Differential topology]], where smooth structures are considered up to diffeomorphism, a type of homeomorphism.
+* [[Knot theory]], where knots are studied as embeddings of circles in 3-dimensional space, up to ambient isotopy, a form of homeomorphism.
+== Related pages ==
+* [[Topological space]]
 * [[Continuous function]]
-* [[Topological space]]
+* [[Isomorphism]]
-* [[Bijection]]
+* [[Diffeomorphism]]
+* [[Knot theory]]
 [[Category:Topology]]
-[[Category:Mathematical concepts]]
-{{Mathematics-stub}}