Knowledge (XXG)

Fixed-point space

Source 📝

238: 71: 130: 95: 147:
is not a fixed-point space, because the continuous function that adds one to its argument does not have a fixed point. Generalizing the unit interval, by the
166:
The definition of a fixed-point space can also be extended from continuous functions of topological spaces to other classes of maps on other types of space.
279: 313: 298: 209: 272: 308: 148: 265: 140: 74: 245: 44: 136: 39: 35: 303: 205: 249: 197: 193: 100: 219: 215: 160: 24: 80: 292: 152: 20: 201: 156: 144: 237: 192:, Springer Monographs in Mathematics, New York: Springer-Verlag, p.  253: 103: 83: 47: 139:is a fixed point space, as can be proved from the 124: 89: 65: 273: 8: 16:Space where all functions have fixed points 280: 266: 188:Granas, Andrzej; Dugundji, James (2003), 102: 82: 46: 175: 7: 234: 232: 183: 181: 179: 252:. You can help Knowledge (XXG) by 14: 236: 66:{\displaystyle f:X\rightarrow X} 113: 107: 57: 1: 314:Mathematical analysis stubs 149:Brouwer fixed-point theorem 38:, according to which every 330: 299:Fixed points (mathematics) 231: 141:intermediate value theorem 202:10.1007/978-0-387-21593-8 163:is a fixed-point space. 248:–related article is a 126: 125:{\displaystyle f(x)=x} 91: 67: 246:mathematical analysis 127: 92: 68: 137:closed unit interval 101: 81: 45: 40:continuous function 36:fixed-point theorem 309:Topological spaces 190:Fixed Point Theory 122: 87: 63: 261: 260: 135:For example, the 90:{\displaystyle x} 32:fixed-point space 321: 282: 275: 268: 240: 233: 223: 222: 185: 131: 129: 128: 123: 96: 94: 93: 88: 72: 70: 69: 64: 329: 328: 324: 323: 322: 320: 319: 318: 289: 288: 287: 286: 229: 227: 226: 212: 187: 186: 177: 172: 161:Euclidean space 99: 98: 79: 78: 43: 42: 25:Hausdorff space 17: 12: 11: 5: 327: 325: 317: 316: 311: 306: 301: 291: 290: 285: 284: 277: 270: 262: 259: 258: 241: 225: 224: 210: 174: 173: 171: 168: 121: 118: 115: 112: 109: 106: 86: 62: 59: 56: 53: 50: 34:if it obeys a 15: 13: 10: 9: 6: 4: 3: 2: 326: 315: 312: 310: 307: 305: 302: 300: 297: 296: 294: 283: 278: 276: 271: 269: 264: 263: 257: 255: 251: 247: 242: 239: 235: 230: 221: 217: 213: 211:0-387-00173-5 207: 203: 199: 195: 191: 184: 182: 180: 176: 169: 167: 164: 162: 158: 154: 150: 146: 142: 138: 133: 119: 116: 110: 104: 84: 76: 60: 54: 51: 48: 41: 37: 33: 29: 26: 22: 254:expanding it 243: 228: 189: 165: 134: 31: 30:is called a 27: 18: 75:fixed point 21:mathematics 293:Categories 170:References 157:convex set 97:for which 77:, a point 145:real line 58:→ 304:Topology 155:bounded 151:, every 220:1987179 153:compact 218:  208:  143:. The 73:has a 244:This 159:in a 250:stub 206:ISBN 23:, a 198:doi 19:In 295:: 216:MR 214:, 204:, 196:, 178:^ 132:. 281:e 274:t 267:v 256:. 200:: 194:2 120:x 117:= 114:) 111:x 108:( 105:f 85:x 61:X 55:X 52:: 49:f 28:X

Index

mathematics
Hausdorff space
fixed-point theorem
continuous function
fixed point
closed unit interval
intermediate value theorem
real line
Brouwer fixed-point theorem
compact
convex set
Euclidean space



2
doi
10.1007/978-0-387-21593-8
ISBN
0-387-00173-5
MR
1987179
Stub icon
mathematical analysis
stub
expanding it
v
t
e
Categories

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.