In the local universe, a large fraction of the baryon content is believed to exist as diffuse gas in filaments. While this gas is directly observable in X-ray emission around clusters of galaxies, it is primarily studied through its UV absorption. Recently, X-ray observations of large-scale filaments connecting to the cosmic web around the nearby ($z=0.05584$) cluster Abell 133 were reported. One of these filaments is intersected by the sightline to quasar [VV98] J010250.2$-$220929, allowing for a first-ever census of cold, cool, and warm gas in a filament of the cosmic web where hot gas has been seen in X-ray emission. Here, we present UV observations with the Cosmic Origins Spectrograph and optical observations with the Magellan Echellette spectrograph of [VV98] J010250.2$-$220929. We find no evidence of cold, cool, or warm gas associated with the filament. In particular, we set a $2sigma$ upper limit on Ly$alpha$ absorption of $log(N_{HI} / textrm{cm}^{-2}) < 13.7$, assuming a Doppler parameter of $b=20,textrm{km},textrm{s}^{-1}$. As this sightline is ${sim}1100,textrm{pkpc}$ ($0.7R_textrm{vir}$) from the center of Abell 133, we suggest that all gas in the filament is hot at this location, or that any warm, cool, or cold components are small and clumpy. A broader census of this system -- combining more UV sightlines, deeper X-ray observations, and a larger redshift catalog of cluster members -- is needed to better understand the roles of filaments around clusters.
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