A zero-dimensional Hausdorff space is inherently unconnected, whereas the reverse is not true. A locally compact Hausdorff space, on **the other hand**, is zero-dimensional if and only if it is completely unconnected. Cantor and Baire spaces are two examples of such spaces.

Properties of spaces with **a small inductive dimension** of zero A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. In particular, a separable metrizable space is zero-dimensional.

A topological space is said to be zero-dimensional (or 0-dimensional) if it is a non-empty T1-space with a base made up of closed sets, that is, sets that are both closed and open at the same time. Topological spaces with zero dimensions are a well-studied class. There are several ways to construct zero-dimensional spaces. The most obvious one is to start with **any set X** and define a new set Y as follows: $$Y = \{0\} \cup X$$ Then $Y$ is trivially zero-dimensional since it has a base consisting of the single set $\{0\}$. More generally, given any non-empty collection ${\mathcal B}$ of subsets of a set X, we can form the union of **all elements** of B. This procedure will yield again a zero-dimensional space since the empty set is closed under arbitrary unions.

Here is another example which will be useful later on: Let X be any set and let $d \in {\mathbb N}\setminus \{0\}$. We can define a new set $X_d$ as follows: $$X_d = \{0, 1, 2, \ldots, d-1\} \times X$$ It is not hard to see that $X_d$ is also zero-dimensional.

Because every open cover in the space has a refinement consisting of **a single open set**, a point is zero-dimensional with regard to the covering dimension. In particular, this implies that every separable metric space is zero-dimensional.

Dimensionless Because it has **no length**, breadth, or height, a point is a zero-dimensional object. It has no dimensions. A line is a one-dimensional object. It has length but no width or depth. A plane is a two-dimensional object. It has length and breadth, but no height.

Three-dimensional An object has three dimensions if it can be placed in space so that any two dimensions are not equal. For example, a box has length, breadth, and height; a cube has length, length, breadth, and height. A box is two-dimensional, while a cube is three-dimensional.

Zero-dimensional objects have no dimension. They cannot be placed in space so that some parts are near others. One-dimensional objects have a length but no breadth. Lines are one-dimensional. Objects with **two or more lines** that don't intersect are examples of **one-dimensional objects**. Two-dimensional objects have length and breadth. Planes have length, breadth, and height. Three-dimensional objects have length, breadth, and height. Zero-dimensional objects are things like points, which cannot be placed even in space, lines which cannot be folded over on themselves, and planes which cannot be crumpled up into a ball.

There are **negative dimensions**, just as there are positive and fractional dimensions. A physicist would say that a shape with negative dimensions could not exist in reality.

However, a mathematician might say that such a thing is possible but leads to some problematic results. For example, if a circle had a negative diameter, then nothing would be able to fit inside it.

Thus, the only real restriction on dimensions is that they must be either positive or zero. There are no other restrictions. A plane with **negative area** is possible, but it wouldn't do anyone **any good**.

Absolute zero, on the other hand, is nearly absolute stillness. As far as we know, nothing in the cosmos or in a laboratory has ever hit absolute zero. Space itself has a temperature of **2.7 kelvins**. However, we now know a specific temperature for it: -459.67 degrees Fahrenheit or -273.15 degrees Celsius, both of which equal 0 kelvin.