T1 not implies US

This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., T1 space (?)) need not satisfy the second topological space property (i.e., US-space (?))
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Statement

A T1 space (i.e., a topological space in which all points are closed) need not be a US-space.

Definitions used

T1 space

Further information: T1 space

A topological space is termed a $T_{1}$-space if every point is closed.

US-space

Further information: US-space

A topological space is termed a US-space if every convergent sequence has a limit.

Related facts

• Hausdorff implies US
• US implies T1
• US not implies Hausdorff

Proof

Example of line with two origins

Further information: line with two origins

Consider the line with two origins -- this is like the real line, except that there are two copies of the origin. Equivalently, it is the quotient of the union of two copies of the real line by the identification of all the nonzero points of one line with the corresponding point of the other line.

This is a $T_{1}$-space, as can be readily checked. It is not a US-space, because a sequence of points approaching the origin is convergent and has two limits: the two origins.