Spacebook Spacebook is a searchable database of topological spaces and their properties (in ZFC) inspired by Steen and Seebach’s Counterexamples in Topology. (Learn more.) Edit Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional Is Is Not None Selected Arcwise Connected Biconnected Cardinality Equal Continuum Cardinality Less Than Continuum Cardinality Not Exceeding Power Set of Continuum Compact Completely Normal Completely Regular Connected Countable Countable Chain Condition Countably Compact Countably Metacompact Countably Paracompact Discrete Extremally Disconnected First Countable Fully Normal Fully T4 Has Dispersion Point Hyperconnected Lindelof Locally Arcwise Connected Locally Compact Locally Connected Locally Pathwise Connected Metacompact Metrizable Normal Paracompact Pathwise Connected Perfectly Normal Pseudocompact Regular Scattered Second Category Second Countable Semiregular Separable Sequentially Compact Sigma Compact Sigma Locally Compact Sigma Locally Finite Base Strong Locally Compact Strongly Connected T0 T1 T2 T2.5 T3 T3.5 T4 T5 Topologically Complete Totally Disconnected Totally Path Disconnected Totally Separated Ultraconnected Urysohn Weakly Countably Compact Zero Dimensional