Space and time are usually confused with length and duration. That is, the physical units of measure that are typically used with space and time are confused with the pure abstractions of space and time.
Let’s call space and time without physical units abstract space and time. Call space and time with physical units concrete space and time. The distinction is between the use of physical measures, which are after all conventions, and abstractions of space and time, which do not use physical measures. Note that an abstract metric for space or time is also an abstraction, not a physical measure.
There are two kinds of concrete space and time, corresponding to the two kinds of physical units of space and time: length and duration. That is, abstract space may be measured by units of length or duration; abstract time may be measured by units of duration or length.
While it is possible to measure both space and time with either length or duration, this introduces a confusion about what it is that length or duration are measuring. Accordingly, it is better to use length to measure either space or time, and duration to measure whichever remains, time or space.
Abstract space is a three dimensional metric or vector space without physical units. Abstract time is a linear order of events or states of space without physical units. These are the domain of mathematics more than physics.
A metric space or vector space can be used to formalize abstract space. A real number line can be used to formalize abstract time. The introduction of physical units does not change the formulation, but one must be careful then to distinguish space and time by their units of measure.
Abstract space with units of length is length space. Abstract time with units of duration is duration time. Since these combinations are the common conventions, they may be abbreviated to space and time if there is no confusion.
Abstract space with units of duration is duration space. Note that duration space has no length; it is extension measured by duration only. Abstract time with units of length is length time. Note that length time has no duration; it is order measured by length only.