Accurate photometric redshift (photo-$z$) estimates are essential to the cosmological science goals of the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST). In this work we use simulated photometry for mock galaxy catalogs to explore how LSST photo-$z$ estimates can be improved by the addition of near-infrared (NIR) and/or ultraviolet (UV) photometry from the Euclid, WFIRST, and/or CASTOR space telescopes. Generally, we find that deeper optical photometry can reduce the standard deviation of the photo-$z$ estimates more than adding NIR or UV filters, but that additional filters are the only way to significantly lower the fraction of galaxies with catastrophically under- or over-estimated photo-$z$. For Euclid, we find that the addition of ${JH}$ $5{sigma}$ photometric detections can reduce the standard deviation for galaxies with $z>1$ ($z>0.3$) by ${sim}20%$ (${sim}10%$), and the fraction of outliers by ${sim}40%$ (${sim}25%$). For WFIRST, we show how the addition of deep ${YJHK}$ photometry could reduce the standard deviation by ${gtrsim}50%$ at $z>1.5$ and drastically reduce the fraction of outliers to just ${sim}2%$ overall. For CASTOR, we find that the addition of its ${UV}$ and $u$-band photometry could reduce the standard deviation by ${sim}30%$ and the fraction of outliers by ${sim}50%$ for galaxies with $z<0.5$. We also evaluate the photo-$z$ results within sky areas that overlap with both the NIR and UV surveys, and when spectroscopic training sets built from the surveys small-area deep fields are used.
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