NORMAL SPACE Meaning and
Definition
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Normal space refers to a mathematical concept used in geometry and topology to describe a specific type of topological space. In this context, a normal space is a topological space that satisfies a particular separation axiom known as the T4 axiom, which is stronger than the T2 or Hausdorff axiom.
The T4 axiom ensures that given any two distinct points in a normal space, there exist disjoint open sets that contain each of the points, respectively. In other words, there is sufficient separation between different points in a normal space.
This type of separation is particularly useful when studying certain properties of spaces and their topological structures. Normal spaces are often associated with familiar geometric shapes, such as Euclidean spaces, and they play a fundamental role in various areas of mathematics, including analysis, topology, and set theory.
The normality property has several implications and consequences. For example, in a normal space, compact sets are closed, and two disjoint compact sets can be separated by disjoint open neighborhoods. Additionally, normality is a necessary condition for several important theorems and concepts, including the Urysohn lemma and the Tietze extension theorem.
Overall, a normal space is a well-behaved topological space that satisfies the T4 separation axiom, ensuring the existence of sufficient separation between distinct points.
Common Misspellings for NORMAL SPACE
- bormal space
- mormal space
- jormal space
- hormal space
- nirmal space
- nkrmal space
- nlrmal space
- nprmal space
- n0rmal space
- n9rmal space
- noemal space
- nodmal space
- nofmal space
- notmal space
- no5mal space
- no4mal space
- nornal space
- norkal space
- norjal space
- normzl space
Etymology of NORMAL SPACE
The etymology of the word "normal space" can be understood by examining the origins of the two constituent terms: "normal" and "space".
1. "Normal":
- The word "normal" originated from the Latin noun "norma", which means a carpenter's square or a rule, indicating the concept of straightness, regularity, or conformity to a standard.
- In mathematics, the term "normal" is often used to describe perpendicularity or orthogonality, indicating a right angle or a relationship that is perpendicular to a given object.
- It evolved in English to mean something that is standard, regular, or conforming to a norm.
2. "Space":
- The word "space" comes from the Latin noun "spatium", referring to room, area, or extent.
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