API gravity
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In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent. We adopt the convention the countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound.
This is called the "countable chain condition" rather than the more logical term "countable antichain condition" for historical reasons related to complete Boolean algebras. (If κ is a cardinal, then in a complete Boolean algebra every antichain has size less than κ if and only if there is no descending κ-sequence of elements, so chain conditions are equivalent to antichain conditions.)
Partial orders and spaces satisfying the ccc are used in the statement of Martin's axiom.
In the theory of forcing, ccc partial orders are used because forcing with any generic set over such an order preserves cardinals and cofinalities.
More generally, if κ is a cardinal then a poset is said to satisfy the κ-chain condition if every antichain has size less than κ. The countable chain condition is the ℵ1-chain condition.
Examples and properties in topology
A topological space is said to satisfy the countable chain condition, or Suslin's Condition, if the partially ordered set of non-empty open subsets of X satisfies the countable chain condition, i.e. every pairwise disjoint collection of non-empty open subsets of X is countable. The name originates from Suslin's Problem.
- Every separable topological space is ccc. Furthermore, the product space of at most c separable spaces is a separable space and, thus, ccc.
- Every metric space is ccc if and only if it's separable, but in general a ccc topological space need not be separable.
For example,
with the product topology is ccc but not separable.
References
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Products of Separable Spaces, K. A. Ross, and A. H. Stone. The American Mathematical Monthly 71(4):pp. 398-403 (1964)