Door space
Encyclopedia
In mathematics
, in the field of topology
, a topological space
is said to be a door space if every subset is either open or closed. The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".
Here are some easy facts about door spaces:
To prove the second assertion, let X be a Hausdorff door space, and let x ≠ y be distinct points. Since X is Hausdorff there are open neighborhoods U and V of x and y respectively such that U∩V=∅. Suppose y is an accumulation point. Then U\{x}∪{y} is closed, since if it were open, then we could say that {y}=(U\{x}∪{y})∩V is open, contradicting that y is an accumulation point. So we conclude that as U\{x}∪{y} is closed, X\(U\{x}∪{y}) is open and hence {x}=U∩[X\(U\{x}∪{y})] is open, implying that x is not an accumulation point.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
is said to be a door space if every subset is either open or closed. The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".
Here are some easy facts about door spaces:
- The discrete spaceDiscrete spaceIn topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
is a door space. - A HausdorffHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
door space has at most one accumulation point. - In a HausdorffHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
door space if x is not an accumulation point then {x} is open.
To prove the second assertion, let X be a Hausdorff door space, and let x ≠ y be distinct points. Since X is Hausdorff there are open neighborhoods U and V of x and y respectively such that U∩V=∅. Suppose y is an accumulation point. Then U\{x}∪{y} is closed, since if it were open, then we could say that {y}=(U\{x}∪{y})∩V is open, contradicting that y is an accumulation point. So we conclude that as U\{x}∪{y} is closed, X\(U\{x}∪{y}) is open and hence {x}=U∩[X\(U\{x}∪{y})] is open, implying that x is not an accumulation point.