Topology

Study of geometric properties preserved under continuous deformations.

What is Topology?Types of Topological SpacesOpen and Closed Sets
Homeomorphisms and Topological EquivalenceConnectedness and CompactnessTopological Invariants
Topology in Computer ScienceTopology in Biology and MedicineEveryday Uses of Topology
Visual Learning with DiagramsActive Recall and Practice Problems
What is Topology?Types of Topological...Open and Closed SetsHomeomorphisms and T...Connectedness and Co...Topological Invarian...Topology in Computer...Topology in Biology ...Everyday Uses of Top...Visual Learning with...Active Recall and Pr...
Advanced Topics

Connectedness and Compactness

How Spaces Stick Together

Topology studies how spaces are connected and how “compact” or “contained” they are.

  • Connectedness: A space is connected if it’s in one piece. If you can split it into two non-overlapping open sets, it’s disconnected.
  • Compactness: A space is compact if every open cover has a finite subcover. In simple terms, it’s contained and doesn’t go off to infinity.

Why These Matter

These properties help mathematicians understand the structure of spaces and solve problems about limits, continuity, and convergence.

See It in Real Life

  • A circle is connected; a pair of separate circles is not.
  • A closed interval \([0, 1]\) on the real line is compact, while the whole real line isn’t.

Examples

  • A piece of string is connected, but two separate strings are not.

  • A closed box is compact; an open-ended tunnel is not.

In a Nutshell

Connectedness and compactness describe how spaces are glued together and how they behave at infinity.

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Topological Invariants
Connectedness and Compactness - Topology Content | Practice Hub