SIMPLY CONNECTED Meaning and
Definition
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Simply connected is a term used in topology and geometry to describe a topological space that possesses a certain property. A topological space is considered simply connected if it is path-connected and every loop in the space can be continuously shrunk to a single point within the space.
To understand this definition more clearly, let's break it down. Firstly, path-connected means that any two points in the space can be connected by a continuous path. In other words, there are no "holes" or disconnected regions within the space.
The second condition, the ability to continuously shrink any loop to a point, means that any closed curve in the space can be gradually deformed or contracted until it becomes a single point while remaining entirely within the space. This implies that any loop can be "untangled" or straightened out without leaving the space.
Alternatively, this can be thought of as the inability to enclose any nontrivial loops within the space. A nontrivial loop would be one that cannot be continuously deformed to a single point without leaving the space.
In essence, a simply connected space is one that does not possess any "holes" and does not allow for the existence of loops that cannot be continuously contracted to a single point within the space. This property has significant implications in various branches of mathematics, physics, and engineering.
Common Misspellings for SIMPLY CONNECTED
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Etymology of SIMPLY CONNECTED
The term "simply connected" originated in the field of topology, which is the study of properties related to spaces and continuous transformations. It was coined in the early 20th century by Henri Poincaré, a French mathematician.
The word "connected" implies that a space cannot be separated into two or more disjoint parts. However, Poincaré introduced the additional concept of simplicity to describe a particular kind of connectedness.
The term "simply" in "simply connected" comes from the French word "simple" meaning "uncomplicated" or "straightforward". Therefore, a simply connected space is one that is not only connected but also free from complicated or nontrivial topological properties.
In simple terms, a space is simply connected if every loop in the space can be continuously contracted to a single point without leaving the space.
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