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In the local universe, a large fraction of the baryon content is believed toexist as diffuse gas in filaments. While this gas is directly observable inX-ray emission around clusters of galaxies, it is primarily studied through itsUV absorption. Recently, X-ray observations of large-scale filaments connectingto the cosmic web around the nearby ($z=0.05584$) cluster Abell 133 werereported. One of these filaments is intersected by the sightline to quasar[VV98] J010250.2$-$220929, allowing for a first-ever census of cold, cool, andwarm gas in a filament of the cosmic web where hot gas has been seen in X-rayemission. Here, we present UV observations with the Cosmic Origins Spectrographand optical observations with the Magellan Echellette spectrograph of [VV98]J010250.2$-$220929. We find no evidence of cold, cool, or warm gas associatedwith the filament. In particular, we set a $2\sigma$ upper limit on Ly$\alpha$absorption of $\log(N_{HI} / \textrm{cm}^{-2}) < 13.7$, assuming a Dopplerparameter 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 anywarm, cool, or cold components are small and clumpy. A broader census of thissystem -- combining more UV sightlines, deeper X-ray observations, and a largerredshift catalog of cluster members -- is needed to better understand the rolesof filaments around clusters.

COS Observations of the Cosmic Web: A Search for the Cooler Components of a Hot, X-ray Identified Filament