The zero space, \(\lbrace0\rbrace\), is a \(0\)-dimensional vector space over every field.

The vector space axioms are satisfied, as vector addition and scalar multiplication become trivial. The basis of the zero space over any field is the empty set \(\lbrace\space\rbrace\).

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A point is infinitely small. It has no width, no depth, no height -- only location. It is a placeholder, a mark of presence. It is a way of saying: there is something here which exists. This is the fundamental barrier between something and nothing -- a jumping off point for universes and infinities, for you and me, for everything that is.

While a \(0\)-dimensional space implies a total disconnectedness of that space, it is possible for a space of \(0\) dimension to be comprised of more than one point. In other words, some spaces can contain multiple points and still be \(0\)-dimensional. Where is the crossing between the disconnection of the zeroth dimension and the continuum of the first dimension? Are your experiences throughout life discrete or do they exhibit an endless continuity? Do you see your place within reality as being along a continuum with your surroundings or as being in isolation from everything around you?

The zero space is often called the trivial space. The notion that something is trivial in mathematics implies that it is obvious, simple, or uninteresting. We say that a solution or a proof is trivial in order to gloss over it and endeavor upon more complex examples and ideas. But the simple can be profound, and it has its own flavor of insight to offer.

The vector space axioms are satisfied, as vector addition and scalar multiplication become trivial. The basis of the zero space over any field is the empty set \(\lbrace\space\rbrace\).

- - - - - - - - - -

A point is infinitely small. It has no width, no depth, no height -- only location. It is a placeholder, a mark of presence. It is a way of saying: there is something here which exists. This is the fundamental barrier between something and nothing -- a jumping off point for universes and infinities, for you and me, for everything that is.

While a \(0\)-dimensional space implies a total disconnectedness of that space, it is possible for a space of \(0\) dimension to be comprised of more than one point. In other words, some spaces can contain multiple points and still be \(0\)-dimensional. Where is the crossing between the disconnection of the zeroth dimension and the continuum of the first dimension? Are your experiences throughout life discrete or do they exhibit an endless continuity? Do you see your place within reality as being along a continuum with your surroundings or as being in isolation from everything around you?

The zero space is often called the trivial space. The notion that something is trivial in mathematics implies that it is obvious, simple, or uninteresting. We say that a solution or a proof is trivial in order to gloss over it and endeavor upon more complex examples and ideas. But the simple can be profound, and it has its own flavor of insight to offer.