**Definitions**

Points *x* and *y* in a topological space *X* can be *separated by neighbourhoods* if there exists a neighbourhood *U* of *x* and a neighbourhood *V* of *y* such that *U* and *V* are disjoint (*U* ∩ *V* = ∅). *X* is a **Hausdorff space** if any two distinct points of *X* can be separated by neighborhoods. This condition is the third separation axiom (after T_{0} and T_{1}), which is why Hausdorff spaces are also called *T _{2} spaces*. The name

*separated space*is also used.

A related, but weaker, notion is that of a **preregular space**. *X* is a preregular space if any two topologically distinguishable points can be separated by neighbourhoods. Preregular spaces are also called *R _{1} spaces*.

The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.

Read more about this topic: Hausdorff Space

### Other articles related to "definition, definitions":

... A theoretical (or conceptual)

**definition**gives the meaning of a word in terms of the theories of a specific discipline ... This type of

**definition**assumes both knowledge and acceptance of the theories that it depends on ... An example of a theoretical

**definition**is that of "Heat" in physics, which actually puts forth an entire theory of heat (involving accelerating molecules, etc.) ...

... Some

**definitions**of language, such as early versions of Charles Hockett's "design features"

**definition**, emphasize the spoken nature of language ... not qualify as a language under these

**definitions**, as it is primarily a written form of communication (to see why, try reading Maxwell's equations out loud) ... However, these

**definitions**would also disqualify sign languages, which are now recognized as languages in their own right, independent of spoken language ...

**Definitions**- South America

... Although topographic maps of the Chile and Argentina border region which contains the highest peaks are of notoriously poor accuracy, with elevation errors exceeding 100 metres (330 ft) in many cases, the current consensus based on the most recent measurements places Ojos del Salado as the 2nd highest peak and highest volcano in South America, significantly higher than Monte Pissis. ...

... the need to have precise terminology that cannot be confused with other words or

**definitions**... on avoidance of words that have many

**definitions**and compared the language of Scientology with the language of Math and other precise doctrines ... Scientology and Dianetics place a heavy emphasis on understanding word

**definitions**...

### Famous quotes containing the word definitions:

“The loosening, for some people, of rigid role *definitions* for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”

—Pamela Patrick Novotny (20th century)

“What I do not like about our *definitions* of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”

—G.C. (Georg Christoph)

“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?

There have been many *definitions* of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”

—Edmond De Goncourt (1822–1896)