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In this article we discuss the following question “When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same ℝ-place?” We use this to show that if R contains a dense real closed subfield R′, then the spaces of ℝ-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X), we prove that its space M(K) of ℝ-places is metrizible if and only if R contains a countable dense subfield. Moreover, we show that this condition is neccessary for the metrizability of M(F) for any function field F of transcendence degree 1 over R.