# Topological Spaces, Metric Spaces and Graphs Answer from bryce. >Topological Spaces, Metric Spaces, and Graphs.  If you squint these all look the same to an extent, but technically they are different. >1. **Topology** is really the concept of 'nearness' >2. A **[Metric](Metric.md)** is a notion of distance,  metrics give us nice topologies (i.e. every metric space is a topological space, but not vice versa). >3. A **graph** is a combinatorial object that can be endowed with a *topology* or a *metric* or both. Given a graph (weighted or not) there is a natural way to think of it as a topological space (abstractly), but not a natural way of thinking of it as a metric space.   There are lots of ways to think about weights on graphs though.  For example, I usually think of them in terms of flow on the graph or resistance.  Pile a bunch of object on each node and let them move down edges based on their weights.  How do those those things flow around the graph, where to things get backed up, where to node become empty.  i.e. Things like the discrete heat equation and or random walks on graphs are often governed by the weights.  You can also use weights to detect the topology of the graph, but they are not inherent to the graph (cochains and cocycles in cohomoology theory).   We can also define a metric as *mathematical object* that converts coordinate intervals to distances. --- Date: 20221228 Links to: Tags: #review References: * []()