What is the inherited topology of a line in RlxR and RlxRl?

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In summary, the topology inherited by a straight line L in the plane as a subspace of RlxR and RlxRl is the union of intervals which is any open interval in R^2. This means that any intersection between open intervals in R^2 and the line will result in an open interval of the line. However, this is not always the case as there are exceptions such as the line y=-x intersecting with the square [a,b)x(c,d) in RlxRl.
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emob2p
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Homework Statement


If L is a straight line in the plane, describe the topology L inherits
as a subspace of RlxR and as a subspace of RlxRl in each case it is a
familiar topology.(Rl= lower limit topology)



The Attempt at a Solution



RlxR topology is the union of intervals [a,b)x(c,d) which is any open interval in R^2. Likewise for the topology of RlxRl. Hence any intersection between open intervals in R^2 and the line y=mx+b will be an open interval of the line. So in both cases, won't the inherited topology just be R?
 
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It won't always be the inherited topology from R. For example in R_l x R, take the line y=-x (i.e. the set {(x,-x)}). What happens when you intersect it with the square [a,b)x(c,d)?
 

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